This is the prayer that is offered by the priest during confession also known as the sacrament of reconciliation or penance.
After one confesses their sins, the priest might give some advice to them and then give them a penance–an act of making amends, usually a few prayers offered to God. They then ask them to say the formal prayer the act of contrition. When that is done they then offer the prayer of absolution which formally forgives their sins ritualistically.
Here are the words, some of the most beautiful in all of Catholic tradition:
God the Father of mercies, through the death and resurrection of your son, you have reconciled the world to yourself and sent the Holy Spirit among us for the forgiveness of sins. Through the ministry of the church, may God grant you pardon and peace. And I absolve you of your sins, in the name of the Father, and of the Son and of the Holy Spirit. Amen.
For more on confession see our two part how-to go to confession video.