The Catholic Church canonized a text reporting that the Risen Lord told his disciples, "If you forgive the sins of any, they are forgiven them; if you retain the sins of any, they are retained." Write a brief "history of forgiveness of sins in the early church," noting views, practices, and conflicts of the church(es) from roughly 100 to 250 CE.

The Catholic Church's view on the forgiveness of sins is encapsulated in the Sacrament of Baptism and the Sacrament of Penance and Reconciliation. The Council of Trent reiterated the Church's stance on penance and forgiveness in 1551; it was little changed from that of earlier centuries.
Thus, from A.D. 100 to 250, the Catholic Church held the view that there were two ways believers could be forgiven for their sins. The first was through baptism into the Church, and the second was through the Sacrament of Penance. The latter was predicated on John 20:23 (as you stated): "If you forgive the sins of any, they are forgiven them; if you retain the sins of any, they are retained."
The early Catholic fathers believed that the power to forgive sins was bequeathed by Christ to his Apostles and, later, to the Roman Catholic clergy. From the earliest centuries, the Church believed that these clergy had power to forgive all types of sin. However, conflict soon arose from different quarters regarding this unlimited power of the Church.
In about A.D. 250, Novatian asserted that the Church had no jurisdiction over heretics or idolaters; he argued that only God could forgive the sins of heresy or idolatry. Similarly, in A.D. 200, Tertullian argued the same: he maintained that some post-baptismal sins were too egregious to be forgiven by mere men. This included the sins of adultery, murder, and apostasy. Tertullian and Novatian were Montanists, and they based their beliefs on Hebrews 6:4-6:

For it is impossible for those who were once enlightened, and have tasted of the heavenly gift, and were made partakers of the Holy Ghost,
And have tasted the good word of God, and the powers of the world to come,
If they shall fall away, to renew them again unto repentance; seeing they crucify to themselves the Son of God afresh, and put him to an open shame.

Others such as Hermas, however, provided that there was one opportunity for forgiveness after a grievous, post-baptismal sin had been committed. The Shepherd of Hermas taught that a first penance was acceptable for such a sin; however, subsequent ones were void under biblical law. To redeem believers who continually lapsed, The Shepherd of Hermas provided for a purgatory (the earliest mention of one), where the offending sinner could be purified through suffering and eventually be made ready for heaven.
Irenaeus of Lyons (from between A.D. 182 and 188) supported Hermas's belief in the forgiveness of egregious sins but disagreed with Tertullian and Novatian's. Irenaeus argued that heretics could be given absolution by clergy and then welcomed back into the Church. Tertullian and Novatian maintained that some sins could never be absolved by the Church alone.
In fact, Tertullian argued that definite steps needed to be taken if one was to be forgiven at all by God.

With regard also to the very dress and food, it commands the penitent to lie in sackcloth and ashes, to cover his body in mourning, to lay his spirit low in sorrows, to exchange for severe treatment the sins which he has committed; moreover, to know no food and drink but such as is plain, not for the stomach's sake, to wit, but the soul's; for the most part, however, to feed prayers on fastings, to groan, to weep and make outcries unto the Lord your God . . . (De Poenitentia by Tertullian).

Meanwhile, Origen (A.D. 185–254) argued that sinners needed to confess their sins to both God and His representatives on earth. He also emphasized the difference between mortal and non-mortal sins. Origen maintained that non-mortal sins could always be forgiven. However, a thorough affliction of the soul and body was required for that to happen. He listed six ways a non-mortal sin could be absolved: through almsgiving, martyrdom, an abundance of love, penitence (by self-torture), forgiveness of another's sin, and bringing another back to the Church.
On the matter of mortal sin, Origen held that only one absolution was possible. For more on the Church fathers' views on the forgiveness of sin from A.D. 100 to 250, please refer to the links provided.