Reconciliation

"Those who approach the sacrament of Penance obtain pardon from God's mercy for the offense committed against him, and are, at the same time, reconciled with the Church which they have wounded by their sins and which by charity, by example, and by prayer labors for their conversion."
Catechism of the Catholic Church, para. 1422

Only God forgives sins. Jesus, who is God, exercised this divine power often during his public life and imparted it upon his Apostles. This is shown in Matthew 16:19: 

“Whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.”

Also, in John 20:21-23: 

"Jesus said to them [the Apostles] again, 'Peace be with you. As the Father has sent me, even so I send you.' And when he had said this, he breathed on them, and said to them, 'Receive the Holy Spirit. If you forgive the sins of any, they are forgiven; if you retain the sins of any, they are retained.'"

The Sacrament of Reconciliation, often referred to as “Confession," allows the penitent to experience God's infinite mercy and forgiveness and obtain pardon for their offenses against Him, and to be reconciled to Him and the whole Church. In the confessional, the priest, who is a successor of the orginal 12 apostles, is in persona Christi: he is acting “in the person of Christ." He is a physical representation of Jesus Christ in the flesh and when you confess your sins, you are confessing them directly to Jesus Himself.