The Catholic Church teaches that man's sins may be absolved by the Church. Absolution from sin means "To free from sin", which is pretty much the same as forgiveness of sins.
In lay-person's terms, as I understand it, the Priest is acting under the authority and power of the Church, which, in turn, is acting under the authority and power of Christ.
This is probably a really poor analogy, but it helps me to keep principles simple, so here goes...
In my household, my oldest child has no authority over the younger ones. She can't tell them what to do under normal circumstances, because she simply has no authority to do so.
However, if I leave her to babysit the younger ones, I give her the authority, and the younger ones understand the principle that she is acting under my authority, and that "to be naughty for her is the same as being naughty for me when I am in charge." But if my oldest tells a younger one to hit another one, or to steal, or do anything I would not authorize, then she would not be acting under the authority I've given her because she is asking them to do something contrary to my will.
Applying the anology...
The way I understand it, the teaching is that the Priest is just a man, not a supernatural being with the power to forgive sins on his own, but when acing under the authority of the Church, which, in turn has been given authority from Christ, and when the Priest is acting in harmony with Christ's instructions, then he has the power to absolve since through the means described and established by the Church in Christ's authority.
The Catholic Encyclopedia says this (small excerpt):
Absolution proper is that act of the priest whereby, in the Sacrament
of Penance, he frees man from sin. It presupposes on the part of the
penitent, contrition, confession, and promise at least of
satisfaction; on the part of the minister, valid reception of the
Order of Priesthood and jurisdiction, granted by competent authority,
over the person receiving the sacrament. That there is in the Church
power to absolve sins committed after baptism the Council of Trent
thus declares: "But the Lord then principally instituted the Sacrament
of Penance, when, being raised from the dead, He breathed upon His
disciples saying, 'Receive ye the Holy Ghost. Whose sins you shall
forgive, they are forgiven them, and whose sins you shall retain, they
are retained.' By which action so signal, and words so clear the
consent of all the Fathers has ever understood that the power of
forgiving and retaining sins was communicated to the Apostles, and to
their lawful successors for the reconciling of the faithful who have
fallen after baptism" (Sess. XIV, i).
