During Sacrament of Penance, as shown in this video, Roman Catholics say, "Bless me, Father, for I have sinned. It's been _ days/months/years since my last Confession."
The penitent is encouraged to engage in a dialogue with the priest, learning what he or she may do in order to avoid sinning next time. Then, the priest will absolve the sin of the penitent, and the penitent is essentially free to go.
In case you get nervous on your first Confession, you may want to plan ahead and recite what you're going to say. Also, remember that the priest cannot really divulge your sins, so that may be a plus to ensure your privacy.
The purpose of the "Bless me, for I have sinned," is that it is supposed to ask the Father to bless the penitent after what the penitent has done, feels sorry for, and wishes to change his/her ways in order to walk in the light of God. The "for I have sinned" part tells the priest that the penitent has sinned and is walking up to the priest for absolution of sin. The "bless me" part is the penitent's request to the priest in order to have the sins absolved. I suppose it may be considered a blessing to be forgiven by God for the sins of the past.
Mark 2:5-12 (English Standard Version)
And when Jesus saw their faith, he said to the paralytic, "Son, your
sins are forgiven. Now some of the scribes were sitting there,
questioning in their hearts, "Why does this man speak like that? He is
blaspheming! Who can forgive sins but God alone?" And immediately
Jesus, perceiving in his spirit that they thus questioned within
themselves, said to them, "Why do you question these things in your
hearts? Which is easier, to say to the paralytic, 'Your sins are
forgiven,' or to say, 'Rise, take up your bed and walk'? But that you
may know that the Son of Man has authority on earth to forgive sins"
--he said to the paralytic-- "I say to you, rise, pick up your bed, and go home."
God can forgive sins, because he's God. Jesus can forgive sins, because in that passage, he says the Son of Man has authority on earth to forgive sins, and well, the gospel portrays him as the Son of Man. Therefore, Jesus can forgive sins. Maybe he's equaling himself to God in the passage, or maybe he wants to promote a view that people should forgive each other, a view later supported by St. Paul in Colossians 3:13 (New International Version):
Bear with each other and forgive one another if any of you has a
grievance against someone. Forgive as the Lord forgave you.
Hope that answers your questions. :)