Hi, Jeff —
Thanks for the question.
Your priest is having a period of Eucharistic Adoration after Mass. This is a great
practice for any parish. It gives anyone, Catholic or non-Catholic, a period where
they can sit, pray, and meditate with Our Lord in the Eucharist.
The star burst shaped relic is a monstrance. It is used to hold Our Blessed
Lord during a period of Adoration of the Blessed Sacrament. Here are some
similar pictures of monstrance's.
After he kneels, he is probably blessing the attendees (those who have stayed after
Mass) with the Blessed Sacrament. The hymn he is singing is sung either in Latin
or English. Latin, because it is the official language of the Church. It is a song
of praise and worship.
Here is a YouTube video of Adoration at a parish.
The thing he is holding in his hands that has smoke coming out from it
is a censer.
It is used for burning incense. The rising smoke represents the prayers, that will be offered by the people for that period, going up to Heaven.
At the end of the period of the Adoration, which will vary from Church
to Church, he will come out again and sing a closing hymn of praise and
place the Eucharist back in the Tabernacle.
Hope this answers your question.