Do Christians have the authority to forgive sin?

I would say to a degree all should (as @curiousdannii said). Take John 20:22-24 into account for this,

If you forgive the sins of any, they are forgiven them; if you withhold forgiveness from any, it is withheld.

I believe what you are asking is more directed to the Office of the Keys as Lutheran's call it (which is more known in Catholicism as Confession/Absolution/Penance). This is not to be confused as a Christian absolving another of sins against others, but an affirmation that with your confession, God hears it, and the Christian conveys that reassurance based on the Gospel (1 John 1:9). Obviously few denominations practice (liturgical ones in addition to Catholicism, like Anglican, Methodist, Orthodox, Lutheran).

The sacrament of confession (or penance or reconcilliation) was not specifically instituted by any of the Gospels. However, all four Gospels address the forgiveness of sin and the sacrament is entirely consistent with what Jesus is purported to have said and done. This is evidenced not only by John 20:22-23, but on the other Gospel accounts of Jesus forgiving sins as an inherent component of the act of healing (Matt 9:2-8; Mark 2:5-12; and Luke 5:20-26).

Early New Testament churches were more focused on compassion, correction, and forgiveness. Mutual correction and forgiveness were part of the fabric of community life. Compassion was balanced with the understanding of the impact of sin upon the communal life and mission of the church. At times, a form of excommunication (1 Cor 5: 3-5; 1 Tim 1: 19-20) was exercised for the good of the church.

Over time, the sacrament of confession changed as the church spread to different cultures and societies. At times, Confession was a sacrament that was to be administered only once during an individual's lifetime. This administration usually was for the forgiveness of particularly onerous offenses such as apostasy, murder of adultery or upon the deathbed of the individual. The sacrament usually required demonstrable proof of repentance and often included a vow of celibacy from that point forward. For this reason, many individuals delayed the sacrament until they were close to death. A penitent (person requesting forgiveness) would be excluded from receiving the Eucharist and would be excused from the celebration at the offertory along with the catechumens.

As the needs of the people and the circumstances of the Church changed, private penance became the norm. By the end of the 6th century, "penance" became known as "confession." The once in a lifetime public rite of penance ended in the Western Church with the Fourth Council of Lateran's decree in 1215 that all baptized Christians must confess their sins and receive Holy Eucharist at least once each year.

The sacrament was influenced over the intervening centuries by many factors. The influence of the Celtic churches was significant. The monastic nature of the Celtic Churches brought the sacrament into a more private environment. In effect, the sacrament was taken out of the church as a whole and into the private chambers of the penitent and his/her confessor (the person hearing the confession). This Celtic influence extended the confessor relationship to priests as well as bishops who had that role exclusively exclusively.

From the eleventh to the fourteenth centuries, major changes occurred in the sacrament of penance. Penance became the satisfaction, confession, contrition and absolution of sins. The emphasis had shifted from the reconciliation of the sinner with the Church (and with God) to doing 'penance.' Also, the act of confession was deemed to have redemptive value in the life of the penitent. That redemptive value also assisted in converting the heart of the sinner, who if truly contrite, was already forgiven even before the act of confession. This aspect contributed to Albigensian and Waldensian heresies that denied the need to confess to a priest. These heresies were condemned by the Fourth Lateran Council as well. The role of the priest was then reinforced and the element of absolution was added to the sacrament.

St. Thomas Acquinas had a great influence on the sacrament and his views were incorporated into the Council of Florence decree for the Armenians in 1439. This document stated that Penance is a sacrament; it consists of contrition of the heart; oral confession to a priest; satisfaction (such as prayer, almsgiving, fasting, etc...); and absolution by the priest. The end result of the sacrament is the forgiveness of sins.

During the Reformation, there were some objections to this formula. Luther accepted Penance as a sacrament, but rejected the aspect that an individual's works were in any way more important than God's Mercy. He also rejected that the power of forgiveness was limited to priests. Calvin accepted private confession and absolution as a means of arousing faith and the recognition of God's Mercy but rejected it as a sacrament. These arguments were addressed by the Council of Trent in 1551. In the Doctrine on the Sacrament of Penance, the following teachings were affirmed: Penance is a sacrament instituted by Christ; it is distinct from Baptism; the three acts of the penitent are contrition, confession of all serious sins and satisfaction; and absolution is reserved to priests alone.

As a result of Vatican II, there has been a revision to the Rite and nature of the sacrament. The purpose of these revisions is to express both the nature and the effect of the sacrament. The purpose is to "obtain pardon from the mercy of God: and to be "reconciled with the Church whom sinners have wounded by their sin." The new Rite of Penance is frequently referred to as the sacrament of Reconciliation. That name more accurately reflects the intended purpose of the sacrament. It takes the focus off the acts of restitution and places it more on the restoration of the individual to the body of Christ, His Church.

There are four forms in the new rite. They are indidual, communal with individual confession and absolution, communal with general absolution and an abbreviated ritual in cases where death is near. The new rite clearly shows the nature and effect of the sacramen; the role of the church community; an emphasis on the reading of God's Word; a shift of emphasis to the public and away from the private forms; and simplicity.

Finally, a thought about the role of the priest in the Sacrament of Penance and Reconciliation. In the Catechism of the Catholic Church, paragraph 1441 states, "Only God forgives sins. Since he is the Son of God, Jesus says of himself, “The Son of man has authority on earth to forgive sins” and exercises this divine power: “Your sins are forgiven.” Further, by virtue of his divine authority he gives this power to men to exercise in his name." This clearly indicates that it is not the individual priest who is forgiving sins and granting absolution... It is the priest acting in the name of Jesus Christ. This is a role that is granted to the priest when Holy Orders are bestowed upon him.

• Fr. Chuck Bradley