Thanks for the question.
Your question can be answered many ways and because 25% of the Catechism of the Catholic Church addresses the topic of Christian Prayer, I have pulling out the three sections I thought would most directly answer your question. If you wish to read everything in context, you can read the whole section on Christian Prayer on-line.
Section One: Prayer In The Christian Life
2558 "Great is the mystery of the faith!" The Church professes this mystery in the Apostles' Creed (Part One) and celebrates it in the sacramental liturgy (Part Two), so that the life of the faithful may be conformed to Christ in the Holy Spirit to the glory of God the Father (Part Three). This mystery, then, requires that the faithful believe in it, that they celebrate it, and that they live from it in a vital and personal relationship with the living and true God. This relationship is prayer.
What Is Prayer?
For me, prayer is a surge of the heart; it is a simple look turned toward heaven,
it is a cry of recognition and of love, embracing both trial and joy.
— St. Thérèse of Lisieux
Prayer as God's gift
2559 "Prayer is the raising of one's mind and heart to God or the requesting of good things from God." (St. John Damascene, Defide orth. 3,24:PG 94,1089C) But when we pray, do we speak from the height of our pride and will, or "out of the depths" of a humble and contrite heart? (Psalm 130:1) He who humbles himself will be exalted; (cf. Luke 18:9-14) humility is the foundation of prayer, Only when we humbly acknowledge that "we do not know how to pray as we ought," (Romans 8:26) are we ready to receive freely the gift of prayer. "Man is a beggar before God." (St. Augustine, Sermo 56,6,9:PL 38,381)
2560 "If you knew the gift of God!" (John 4:10) The wonder of prayer is revealed beside the well where we come seeking water: there, Christ comes to meet every human being. It is he who first seeks us and asks us for a drink. Jesus thirsts; his asking arises from the depths of God's desire for us. Whether we realize it or not, prayer is the encounter of God's thirst with ours. God thirsts that we may thirst for him. (cf. St. Augustine, De diversis quaestionibus octoginta tribus 64,4:PL 40,56.)
2561 "You would have asked him, and he would have given you living water." (John 4:10) Paradoxically our prayer of petition is a response to the plea of the living God: "They have forsaken me, the fountain of living waters, and hewn out cisterns for themselves, broken cisterns that can hold no water!" (Jeremiah 2:13) Prayer is the response of faith to the free promise of salvation and also a response of love to the thirst of the only Son of God. (cf. John 7:37-39; 19:28; Isaiah 12:3; 51:1; Zechariah 12:10; 13:1)
Prayer as covenant
2562 Where does prayer come from? Whether prayer is expressed in words or gestures, it is the whole man who prays. But in naming the source of prayer, Scripture speaks sometimes of the soul or the spirit, but most often of the heart (more than a thousand times). According to Scripture, it is the heart that prays. If our heart is far from God, the words of prayer are in vain.
2563 The heart is the dwelling-place where I am, where I live; according to the Semitic or Biblical expression, the heart is the place "to which I withdraw." The heart is our hidden center, beyond the grasp of our reason and of others; only the Spirit of God can fathom the human heart and know it fully. The heart is the place of decision, deeper than our psychic drives. It is the place of truth, where we choose life or death. It is the place of encounter, because as image of God we live in relation: it is the place of covenant.
2564 Christian prayer is a covenant relationship between God and man in Christ. It is the action of God and of man, springing forth from both the Holy Spirit and ourselves, wholly directed to the Father, in union with the human will of the Son of God made man.
Prayer as communion
2565 In the New Covenant, prayer is the living relationship of the children of God with their Father who is good beyond measure, with his Son Jesus Christ and with the Holy Spirit. The grace of the Kingdom is "the union of the entire holy and royal Trinity . . . with the whole human spirit." (St. Gregory of Nazianzus, Oratio, 16,9:PG 35,945) Thus, the life of prayer is the habit of being in the presence of the thrice-holy God and in communion with him. This communion of life is always possible because, through Baptism, we have already been united with Christ. (cf. Romans 6:5) Prayer is Christian insofar as it is communion with Christ and extends throughout the Church, which is his Body. Its dimensions are those of Christ's love. (cf. Ephesians 3:18-21)
Chapter One - The Revelation of Prayer
Article 3 - In The Age of The Church
2623 On the day of Pentecost, the Spirit of the Promise was poured out on the disciples, gathered "together in one place." (Acts 2:1) While awaiting the Spirit, "all these with one accord devoted themselves to prayer." (Acts 1:14) The Spirit who teaches the Church and recalls for her everything that Jesus said (cf. John 14:26) was also to form her in the life of prayer.
2624 In the first community of Jerusalem, believers "devoted themselves to the apostles' teaching and fellowship, to the breaking of bread, and the prayers." (Acts 2:42) This sequence is characteristic of the Church's prayer: founded on the apostolic faith; authenticated by charity; nourished in the Eucharist.
2625 In the first place these are prayers that the faithful hear and read in the Scriptures, but also that they make their own - especially those of the Psalms, in view of their fulfillment in Christ. (cf. Luke 24:27,44) The Holy Spirit, who thus keeps the memory of Christ alive in his Church at prayer, also leads her toward the fullness of truth and inspires new formulations expressing the unfathomable mystery of Christ at work in his Church's life, sacraments, and mission. These formulations are developed in the great liturgical and spiritual traditions. The forms of prayer revealed in the apostolic and canonical Scriptures remain normative for Christian prayer.
I. Blessing and Adoration
2626 Blessing expresses the basic movement of Christian prayer: it is an encounter between God and man. In blessing, God's gift and man's acceptance of it are united in dialogue with each other. The prayer of blessing is man's response to God's gifts: because God blesses, the human heart can in return bless the One who is the source of every blessing.
2627 Two fundamental forms express this movement: our prayer ascends in the Holy Spirit through Christ to the Father - we bless him for having blessed us; (cf. Ephesians 1:3-14; 2 Corinthians 1:3-7; 1 Peter 1:3-9) it implores the grace of the Holy Spirit that descends through Christ from the Father - he blesses us. (cf. 2 Corinthians 13:14; Romans 15:5-6,13; Ephesians 6:23-24)
2628 Adoration is the first attitude of man acknowledging that he is a creature before his Creator. It exalts the greatness of the Lord who made us (cf. Psalm 95:1-6) and the almighty power of the Savior who sets us free from evil. Adoration is homage of the spirit to the "King of Glory," (Psalm 24, 9-10) respectful silence in the presence of the "ever greater" God. (cf. St. Augustine, En. in Ps. 62,16:PL 36,757-758) Adoration of the thrice-holy and sovereign God of love blends with humility and gives assurance to our supplications.
II. Prayer of Petition
2629 The vocabulary of supplication in the New Testament is rich in shades of meaning: ask, beseech, plead, invoke, entreat, cry out, even "struggle in prayer." (cf. Romans 15:30; Colossians 4:12) Its most usual form, because the most spontaneous, is petition: by prayer of petition we express awareness of our relationship with God. We are creatures who are not our own beginning, not the masters of adversity, not our own last end. We are sinners who as Christians know that we have turned away from our Father. Our petition is already a turning back to him.
2630 The New Testament contains scarcely any prayers of lamentation, so frequent in the Old Testament. In the risen Christ the Church's petition is buoyed by hope, even if we still wait in a state of expectation and must be converted anew every day. Christian petition, what St. Paul calls {"groaning," arises from another depth, that of creation "in labor pains" and that of ourselves "as we wait for the redemption of our bodies. For in this hope we were saved." (Romans 8:22-24) In the end, however, "with sighs too deep for words" the Holy Spirit "helps us in our weakness; for we do not know how to pray as we ought, but the Spirit himself intercedes for us with sighs too deep for words." (Romans 8:26)
2631 The first movement of the prayer of petition is asking forgiveness, like the tax collector in the parable: "God, be merciful to me a sinner!" (Luke 18:13) It is a prerequisite for righteous and pure prayer. A trusting humility brings us back into the light of communion between the Father and his Son Jesus Christ and with one another, so that "we receive from him whatever we ask." (1 John 3:22; cf. 1:7-2:2) Asking forgiveness is the prerequisite for both the Eucharistic liturgy and personal prayer.
2632 Christian petition is centered on the desire and search for the Kingdom to come, in keeping with the teaching of Christ. (cf. Matthew 6:10,33; Luke 11:2,13) There is a hierarchy in these petitions: we pray first for the Kingdom, then for what is necessary to welcome it and cooperate with its coming. This collaboration with the mission of Christ and the Holy Spirit, which is now that of the Church, is the object of the prayer of the apostolic community. (cf. Acts 6:6; 13:3) It is the prayer of Paul, the apostle par excellence, which reveals to us how the divine solicitude for all the churches ought to inspire Christian prayer. (cf. Romans 10:1; Ephesians 1:16-23; Philippians 1:9-11; Colossians 1:3-6; 4:3-4,12) By prayer every baptized person works for the coming of the Kingdom.
2633 When we share in God's saving love, we understand that every need can become the object of petition. Christ, who assumed all things in order to redeem all things, is glorified by what we ask the Father in his name. (cf. John 14:13) It is with this confidence that St. James and St. Paul exhort us to pray at all times. (cf. James 1:5-8; Ephesians 5:20; Philippians 4:6-7; Colossians 3:16-17; 1 Thessalonians 5:17-18)
III. Prayer of Intercession
2634 Intercession is a prayer of petition which leads us to pray as Jesus did. He is the one intercessor with the Father on behalf of all men, especially sinners. (cf. Romans 8:34; 1 John 2:1; 1 Timothy 2:5-8) He is "able for all time to save those who draw near to God through him, since he always lives to make intercession for them." (Hebrews 7:25) The Holy Spirit "himself intercedes for us . . . and intercedes for the saints according to the will of God." (Romans 8:26-27)
2635 Since Abraham, intercession - asking on behalf of another has been characteristic of a heart attuned to God's mercy. In the age of the Church, Christian intercession participates in Christ's, as an expression of the communion of saints. In intercession, he who prays looks "not only to his own interests, but also to the interests of others," even to the point of praying for those who do him harm. (Philippians 2:4; cf. Acts 7:60; Luke 23:28,34)
2636 The first Christian communities lived this form of fellowship intensely. (cf. Acts 12:5; 20:36; 21:5; 2 Corinthians 9:14) Thus the Apostle Paul gives them a share in his ministry of preaching the Gospel (cf. Ephesians 6:18-20; Colossians 4:3-4; 1 Thessalonians 5:25) but also intercedes for them. (cf. 2 Thessalonians 1:11; Colossians 1:3; Philippians 1:3-4) The intercession of Christians recognizes no boundaries: "for all men, for kings and all who are in high positions," for persecutors, for the salvation of those who reject the Gospel. (2 Timothy 2:1; cf. Romans 12:14; 10:1)
IV. Prayer of Thanksgiving
2637 Thanksgiving characterizes the prayer of the Church which, in celebrating the Eucharist, reveals and becomes more fully what she is. Indeed, in the work of salvation, Christ sets creation free from sin and death to consecrate it anew and make it return to the Father, for his glory. The thanksgiving of the members of the Body participates in that of their Head.
2638 As in the prayer of petition, every event and need can become an offering of thanksgiving. The letters of St. Paul often begin and end with thanksgiving, and the Lord Jesus is always present in it: "Give thanks in all circumstances; for this is the will of God in Christ Jesus for you"; "Continue steadfastly in prayer, being watchful in it with thanksgiving." (1 Thessalonians 5:18; Colossians 4:2)
V. Prayer of Praise
2639 Praise is the form of prayer which recognizes most immediately that God is God. It lauds God for his own sake and gives him glory, quite beyond what he does, but simply because HE IS. It shares in the blessed happiness of the pure of heart who love God in faith before seeing him in glory. By praise, the Spirit is joined to our spirits to bear witness that we are children of God, (cf. Romans 8:16) testifying to the only Son in whom we are adopted and by whom we glorify the Father. Praise embraces the other forms of prayer and carries them toward him who is its source and goal: the "one God, the Father, from whom are all things and for whom we exist." (1 Corinthians 8:6)
2640 St. Luke in his gospel often expresses wonder and praise at the marvels of Christ and in his Acts of the Apostles stresses them as actions of the Holy Spirit: the community of Jerusalem, the invalid healed by Peter and John, the crowd that gives glory to God for that, and the pagans of Pisidia who "were glad and glorified the word of God." (Acts 2:47; 3:9; 4:21; 13:48)
2641 "[Address] one another in psalms and hymns and spiritual songs, singing and making melody to the Lord with all your heart." (Ephesians 5:19; Colossians 3:16) Like the inspired writers of the New Testament, the first Christian communities read the Book of Psalms in a new way, singing in it the mystery of Christ. In the newness of the Spirit, they also composed hymns and canticles in the light of the unheard-of event that God accomplished in his Son: his Incarnation, his death which conquered death, his Resurrection, and Ascension to the right hand of the Father. (cf. Philippians 2:6-11; Colossians 1:15-20; Ephesians 5:14; 1 Timothy 3:16; 6:15-16; 2 Timothy 2:11-13) Doxology, the praise of God, arises from this "marvelous work" of the whole economy of salvation. (cf. Ephesians 1:3-14; Romans 16:25-27; Ephesians 3:20-21; Jude 24-25)
2642 The Revelation of "what must soon take place," the Apocalypse, is borne along by the songs of the heavenly liturgy (cf. Revelation 4:8-11; 5:9-14; 7:10-12) but also by the intercession of the "witnesses" (martyrs). (Revelation 6:10) The prophets and the saints, all those who were slain on earth for their witness to Jesus, the vast throng of those who, having come through the great tribulation, have gone before us into the Kingdom, all sing the praise and glory of him who sits on the throne, and of the Lamb. (cf. Revelation 18:24; 19:1-8) In communion with them, the Church on earth also sings these songs with faith in the midst of trial. By means of petition and intercession, faith hopes against all hope and gives thanks to the "Father of lights," from whom "every perfect gift" comes down. (James 1:17) Thus faith is pure praise.
2643 The Eucharist contains and expresses all forms of prayer: it is "the pure offering" of the whole Body of Christ to the glory of God's name (cf. Malachi 1:11) and, according to the traditions of East and West, it is the "sacrifice of praise."
In Brief
2644 The Holy Spirit who teaches the Church and recalls to her all that Jesus said also instructs her in the life of prayer, inspiring new expressions of the same basic forms of prayer: blessing, petition, intercession, thanksgiving, and praise.
2645 Because God blesses the human heart, it can in return bless him who is the source of every blessing.
2646 Forgiveness, the quest for the Kingdom, and every true need are objects of the prayer of petition.
2647 Prayer of intercession consists in asking on behalf of another. It knows no boundaries and extends to one's enemies.
2648 Every joy and suffering, every event and need can become the matter for thanksgiving which, sharing in that of Christ, should fill one's whole life: "Give thanks in all circumstances" (1 Thessalonians 5:18).
2649 Prayer of praise is entirely disinterested and rises to God, lauds him, and gives him glory for his own sake, quite beyond what he has done, but simply because HE IS.
Chapter Three -- The Life of Prayer
Article 1 - Expressions of Prayer
I. Vocal Prayer
2700 Through his Word, God speaks to man. By words, mental or vocal, our prayer takes flesh. Yet it is most important that the heart should be present to him to whom we are speaking in prayer: "Whether or not our prayer is heard depends not on the number of words, but on the fervor of our souls." (St. John Chrysostom, Ecloga de oratione 2:PG 63,585)
2701 Vocal prayer is an essential element of the Christian life. To his disciples, drawn by their Master's silent prayer, Jesus teaches a vocal prayer, the Our Father. He not only prayed aloud the liturgical prayers of the synagogue but, as the Gospels show, he raised his voice to express his personal prayer, from exultant blessing of the Father to the agony of Gethsemane. (cf. Matthew 11:25-26; Mark 14:36)
2702 The need to involve the senses in interior prayer corresponds to a requirement of our human nature. We are body and spirit, and we experience the need to translate our feelings externally. We must pray with our whole being to give all power possible to our supplication.
2703 This need also corresponds to a divine requirement. God seeks worshippers in Spirit and in Truth, and consequently living prayer that rises from the depths of the soul. He also wants the external expression that associates the body with interior prayer, for it renders him that perfect homage which is his due.
2704 Because it is external and so thoroughly human, vocal prayer is the form of prayer most readily accessible to groups. Even interior prayer, however, cannot neglect vocal prayer. Prayer is internalized to the extent that we become aware of him "to whom we speak;" (St. Teresa of Jesus, The Way of Perfection 26,9 in The Collected Works of St. Teresa of Avila, translated. K. Kavanaugh, OCD, and O. Rodriguez, OCD (Washington DC: Institute of Carmelite Studies, 1980),II,136) Thus vocal prayer becomes an initial form of contemplative prayer.
II. Meditation
2705 Meditation is above all a quest. The mind seeks to understand the why and how of the Christian life, in order to adhere and respond to what the Lord is asking. The required attentiveness is difficult to sustain. We are usually helped by books, and Christians do not want for them: the Sacred Scriptures, particularly the Gospels, holy icons, liturgical texts of the day or season, writings of the spiritual fathers, works of spirituality, the great book of creation, and that of history the page on which the "today" of God is written.
2706 To meditate on what we read helps us to make it our own by confronting it with ourselves. Here, another book is opened: the book of life. We pass from thoughts to reality. To the extent that we are humble and faithful, we discover in meditation the movements that stir the heart and we are able to discern them. It is a question of acting truthfully in order to come into the light: "Lord, what do you want me to do?"
2707 There are as many and varied methods of meditation as there are spiritual masters. Christians owe it to themselves to develop the desire to meditate regularly, lest they come to resemble the three first kinds of soil in the parable of the sower. (cf. Mark 4:4-7, 15-19) But a method is only a guide; the important thing is to advance, with the Holy Spirit, along the one way of prayer: Christ Jesus.
2708 Meditation engages thought, imagination, emotion, and desire. This mobilization of faculties is necessary in order to deepen our convictions of faith, prompt the conversion of our heart, and strengthen our will to follow Christ. Christian prayer tries above all to meditate on the mysteries of Christ, as in Lectio Divina or the Rosary. This form of prayerful reflection is of great value, but Christian prayer should go further: to the knowledge of the love of the Lord Jesus, to union with him.
II. Contemplative Prayer
2709 What is contemplative prayer? St. Teresa answers: "Contemplative prayer [oracion mental] in my opinion is nothing else than a close sharing between friends; it means taking time frequently to be alone with him who we know loves us." (St. Teresa of Jesus, The Book of Her Life, 8,5 in The Collected Works of St. Teresa of Avila, translated K. Kavanaugh, OCD, and O. Rodriguez, OCD (Washington DC: Institute of Carmelite Studies, 1976),I,67) Contemplative prayer seeks him "whom my soul loves." (Song of Songs 1:7; cf. 3:14) It is Jesus, and in him, the Father. We seek him, because to desire him is always the beginning of love, and we seek him in that pure faith which causes us to be born of him and to live in him. In this inner prayer we can still meditate, but our attention is fixed on the Lord himself.
2710 The choice of the time and duration of the prayer arises from a determined will, revealing the secrets of the heart. One does not undertake contemplative prayer only when one has the time: one makes time for the Lord, with the firm determination not to give up, no matter what trials and dryness one may encounter. One cannot always meditate, but one can always enter into inner prayer, independently of the conditions of health, work, or emotional state. The heart is the place of this quest and encounter, in poverty and in faith.
2711 Entering into contemplative prayer is like entering into the Eucharistic liturgy: we "gather up:" the heart, recollect our whole being under the prompting of the Holy Spirit, abide in the dwelling place of the Lord which we are, awaken our faith in order to enter into the presence of him who awaits us. We let our masks fall and turn our hearts back to the Lord who loves us, so as to hand ourselves over to him as an offering to be purified and transformed.
2712 Contemplative prayer is the prayer of the child of God, of the forgiven sinner who agrees to welcome the love by which he is loved and who wants to respond to it by loving even more. (cf. Luke 7:36-50; 19:1-10) But he knows that the love he is returning is poured out by the Spirit in his heart, for everything is grace from God. Contemplative prayer is the poor and humble surrender to the loving will of the Father in ever deeper union with his beloved Son.
2713 Contemplative prayer is the simplest expression of the mystery of prayer. It is a gift, a grace; it can be accepted only in humility and poverty. Contemplative prayer is a covenant relationship established by God within our hearts. (cf. Jeremiah 31:33) Contemplative prayer is a communion in which the Holy Trinity conforms man, the image of God, "to his likeness."
2714 Contemplative prayer is also the pre-eminently intense time of prayer. In it the Father strengthens our inner being with power through his Spirit "that Christ may dwell in [our] hearts through faith" and we may be "grounded in love." (Ephesians 3:16-17)
2715 Contemplation is a gaze of faith, fixed on Jesus.
"I look at him and he looks at me": this is what a certain peasant of Ars in the time of his holy curé used to say while praying before the tabernacle. This focus on Jesus is a renunciation of self. His gaze purifies our heart; the light of the countenance of Jesus illumines the eyes of our heart and teaches us to see everything in the light of his truth and his compassion for all men. Contemplation also turns its gaze on the mysteries of the life of Christ. Thus it learns the "interior knowledge of our Lord," the more to love him and follow him.
(cf. St. Ignatius of Loyola, Spiritual Exercises, 104)
2716 Contemplative prayer is hearing the Word of God. Far from being passive, such attentiveness is the obedience of faith, the unconditional acceptance of a servant, and the loving commitment of a child. It participates in the "Yes" of the Son become servant and the Fiat of God's lowly handmaid.
2717 Contemplative prayer is silence, the "symbol of the world to come" (cf. St. Isaac of Nineveh, Tract. myst. 66) or "silent love." (St. John of the Cross, Maxims and Counsels, 53 in The Collected Works of St. John of the Cross, translated K. Kavanaugh, OCD, and O. Rodriguez, OCD (Washington DC: Institute of Carmelite Studies, 1979), 678) Words in this kind of prayer are not speeches; they are like kindling that feeds the fire of love. In this silence, unbearable to the "outer" man, the Father speaks to us his incarnate Word, who suffered, died, and rose; in this silence the Spirit of adoption enables us to share in the prayer of Jesus.
2718 Contemplative prayer is a union with the prayer of Christ insofar as it makes us participate in his mystery. The mystery of Christ is celebrated by the Church in the Eucharist, and the Holy Spirit makes it come alive in contemplative prayer so that our charity will manifest it in our acts.
2719 Contemplative prayer is a communion of love bearing Life for the multitude, to the extent that it consents to abide in the night of faith. The Paschal night of the Resurrection passes through the night of the agony and the tomb - the three intense moments of the Hour of Jesus which his Spirit (and not "the flesh [which] is weak") brings to life in prayer. We must be willing to "keep watch with [him] one hour." (cf. Matthew 26:40)
In Brief
2720 The Church invites the faithful to regular prayer: daily prayers, the Liturgy of the Hours, Sunday Eucharist, the feasts of the liturgical year.
2721 The Christian tradition comprises three major expressions of the life of prayer: vocal prayer, meditation, and contemplative prayer. They have in common the recollection of the heart.
2722 Vocal prayer, founded on the union of body and soul in human nature, associates the body with the interior prayer of the heart, following Christ's example of praying to his Father and teaching the Our Father to his disciples.
2723 Meditation is a prayerful quest engaging thought, imagination, emotion, and desire. Its goal is to make our own in faith the subject considered, by confronting it with the reality of our own life.
2724 Contemplative prayer is the simple expression of the mystery of prayer. It is a gaze of faith fixed on Jesus, an attentiveness to the Word of God, a silent love. It achieves real union with the prayer of Christ to the extent that it makes us share in his mystery.
This better not be for a test or term paper, if so, make sure you give the appropriate source for your information in the notes.
