St. Callixtus was already a confessor when he became pope; he had suffered for the faith, being sentenced to hard labor, and near death, in the mines for being a Christian. When he became pope, he began a program of actively allowing converts from other sects to participate in church life without additional penance; according to Hippolytus, if I understand him correctly, he took the conversion itself to be sign of repentance and declared that they were absolved personally by himself. As St. Hippolytus puts it in his attack on St. Callixtus in Refutation of All Heresies, Book 9, Chapter 7:
And he first invented the device of conniving with men in regard of their indulgence in sensual pleasures, saying that all had their sins forgiven by himself. For he who is in the habit of attending the congregation of any one else, and is called a Christian, should he commit any transgression; the sin, they say, is not reckoned unto him, provided only he hurries off and attaches himself to the school of Callistus. And many persons were gratified with his regulation, as being stricken in conscience, and at the same time having been rejected by numerous sects; while also some of them, in accordance with our condemnatory sentence, had been by us forcibly ejected from the Church. Now such disciples as these passed over to these followers of Callistus, and served to crowd his school. This one propounded the opinion, that, if a bishop was guilty of any sin, if even a sin unto death, he ought not to be deposed. About the time of this man, bishops, priests, and deacons, who had been twice married, and thrice married, began to be allowed to retain their place among the clergy. If also, however, any one who is in holy orders should become married, Callistus permitted such a one to continue in holy orders as if he had not sinned. And in justification, he alleges that what has been spoken by the Apostle has been declared in reference to this person: "Who are you that judgest another man's servant? " But he asserted that likewise the parable of the tares is uttered in reference to this one: "Let the tares grow along with the wheat;" or, in other words, let those who in the Church are guilty of sin remain in it. But also he affirmed that the ark of Noe was made for a symbol of the Church, in which were both dogs, and wolves, and ravens, and all things clean and unclean; and so he alleges that the case should stand in like manner with the Church.
So, in other words, St. Callixtus allowed easy conversions and married clergy, and when confronted with this, replied, "Who are you to judge?" St. Hippolytus goes on to argue that this attitude led to all sorts of nonsense: people, instead of repenting, justified their sins, people who were not legally married counted themselves as married, and women began to use contraception and drugs for abortions extensively. It was, St. Hippolytus thought, a terrible thing masquerading as the Catholic Church.
A bunch of people who agreed with St. Hippolytus elected him pope -- it is unclear whether it was during St. Callixtus's tenure or at the end of it, although Hippolytus himself does make it sound as if it were during. The problem was that the person who was actually elected pope after Callixtus, in a way following the traditional customs of Rome, was St. Urban I. St. Hippolytus and his followers had split the church. We know almost nothing about St. Urban I, beyond his dates (he is the first pope whose reign can be precisely dated using independent historical sources) and the fact that there were many converts during his reign. He may have been martyred -- but even this is unclear. What we do know is that St. Hippolytus's schism continued through his papacy. After St. Urban came St. Pontian; and St. Hippolytus's schism continued. We know only bits and pieces of Pontian's reign, but we do know that almost five years of relative peace were shattered by the rise of Maximinus to the Imperial throne. Christians began to rounded up quite aggressively. St. Pontian, realizing as the nets began to close that he would not be able to avoid capture, did something that no pope had ever done before: on the 28th of September, AD 235, he resigned, in order to guarantee an orderly succession. He was arrested almost immediately afterward, and St. Anterus was elected the next pope.
St. Hippolytus had also been caught in the persecution, and both St. Pontian and St. Hippolytus were sentenced to heavy labor in the mines of Sardinia, which was for all practical purposes a death sentence. Both St. Pontian and St. Hippolytus died there.
St. Anterus was martyred and succeeded by St. Fabian. At this point the persecution was letting up, and thus St. Fabian was able to reclaim the bodies of martyrs and bring them back to Rome for proper burial. He brought back the bodies of both St. Pontian and St. Hippolytus, buried them both with full honors as martyrs, and put them both on the calendar. As far as we can tell, it ended the schism completely.
A lot of stories say that St. Pontian and St. Hippolytus reconciled in the mines of Sardinia. That would be wonderful, but in reality we have no definite reason to believe this. For all we know, they were opponents till the end. But they were both martyrs. They were both wise and holy men. They are both on the calendar of saints. And they both share the same feastday, today, the 13th of August; as is only fitting.