After having confessed your sins, your confessor will probably offer some advice or maybe even ask a few more questions to help open your heart to the healing grace of this sacrament. Then, he will invite you to make an act of contrition before he says the prayer of absolution.
Traditionally, the Act of Contrition was a set prayer that everyone was urged to memorize and recite in the confessional. A form of that prayer is given below. But with the revision of the sacrament after Vatican II, the option was given for penitents to offer their own, more personal, prayer in its place. Any wording is acceptable, provided it includes a statement of true sorrow for our sins and a statement of resolution to do our best to avoid sin in the future. Below is a traditional act of contrition.
O my God, I am heartily sorry for having offended you, and I detest all my sins, because of your just punishments, but most of all because they offend you, my God, who are all-good and deserving of all my love. I firmly resolve, with the help of your grace, to sin no more and to avoid the near occasion of sin.