The sacrament of penance consists of confessing one's sins to a priest and receiving God's forgiveness through the priest. This is an almost uniquely Catholic practice, but it is one of great consolation and relief. When we go to confession and hear the words of absolution from the priest, we can know that God has forgiven our sins.
Institution in the Bible
After His resurrection, Jesus gave His apostles the authority to forgive sins. “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.'” - John 20: 22 – 23. This is a very clear instruction from Christ that the Apostles have the power to forgive or not forgive sins.
Early Church Fathers Confession existed
In the early days of the Church, the sacrament of penance was widely used. They understood it to mean confessing sins to the priest, and receiving forgiveness through him from God.
[Regarding confession, some] flee from this work as being an exposure of themselves, or they put it off from day to day. I presume they are more mindful of modesty than of salvation, like those who contract a disease in the more shameful parts of the body and shun making themselves known to the physicians; and thus they perish along with their own bashfulness. - Tertullian Repentance 10:1
[A final method of forgiveness], albeit hard and laborious [is] the remission of sins through penance, when the sinner...does not shrink from declaring his sin to a priest of the Lord and from seeking medicine, after the manner of him who say, ‘I said, "To the Lord I will accuse myself of my iniquity"’. - Origen Homilies on Leviticus 2:4
It is necessary to confess our sins to those to whom the dispensation of God’s mysteries is entrusted. Those doing penance of old are found to have done it before the saints. It is written in the Gospel that they confessed their sins to John the Baptist (Matt. 3:6), but in Acts (19:18) they confessed to the apostles. - Basil the Great Rules Briefly Treated 288
For those to whom [the right of binding and loosing] has been given, it is plain that either both are allowed, or it is clear that neither is allowed. Both are allowed to the Church, neither is allowed to heresy. For this right has been granted to priests only. - Ambrose Penance 1:1
If the serpent, the devil, bites someone secretly, he infects that person with the venom of sin. And if the one who has been bitten keeps silence and does not do penance, and does not want to confess his wound...then his brother and his master, who have the word [of absolution] that will cure him, cannot very well assist him. - Jerome Commentary on Ecclesiastes 10:11
Father who knowest the hearts of all grant upon this Thy servant whom Thou hast chosen for the episcopate to feed Thy holy flock and serve as Thine high priest, that he may minister blamelessly by night and day, that he may unceasingly behold and appropriate Thy countenance and offer to Thee the gifts of Thy holy Church. And that by the high priestly Spirit he may have authority to forgive sins. - Hippolytus Apostolic Tradition 3
All mortal sins are to be submitted to the keys of the Church and all can be forgiven; but recourse to these keys is the only, the necessary, and the certain way to forgiveness. Unless those who are guilty of grievous sin have recourse to the power of the keys, they cannot hope for eternal salvation. Open your lips, them, and confess your sins to the priest. Confession alone is the true gate to Heaven. - Augustine Christian Combat
These early Christian authors all speak of the necessity of confessing one's sins to a priest. It is the only way which Christ gave us to be sure that our sins would be forgiven.