For complete religion and church guide, see here: Religion & Keeping the Church Strong.
The pope has several powers. The first, and strongest, is crusades. The pope can call a crusade as long as the moral authority of the catholic church is 5% or higher. This calls for a religious war, generally against muslims, but after 1198 crusades can also be called upon pagans. The goal of a crusade is either to take an area considered holy by the catholic church, or to take back land taken from christians by another religious group.
Second, but also importantly, the pope has the power to excommunicate catholic rulers as long as the moral authority of the church is above 20%. This will make the ruler's subjects get a major opinion penalty of -30 towards the ruler, and will give everyone casus belli against the excommunicated ruler This costs the person who asked for excommunication 100 piety.
Third, the pope can condone invasions. More detail on this in the installment on casus belli. This can be done as long as the MA of the church is above 30%
Fourth, the pope can conduct divorces. Short of one part of the couple dying, this is the only way to end a marriage. This costs the person who asked for the divorce 100 piety, and will make the spouse's family unhappy.
Fifth, all church taxes are paid to the pope unless the bishop prefers his liege or an antipope. This can be a huge amount of money per year, and as such the Papacy is usually one of the richest realms in the game.
On the other hand, there are also antipopes. Antipopes are bishops who directly challenge the popes authority, and they are a constant drain on the moral authority of the church. An antipope has two of the powers that popes have; excommunication and taxation.
The character who controls an antipope will be able to request excommunication of anyone in his realm, and he and all his vassals will be protected from excommunication by the real pope. This can be a major asset if the current pope doesn't like you, and you have political enemies with high enough piety, and good enough graces with the pope, to get you excommunicated.
Second, an antipope also has the potential to bring in huge amounts of taxes. Every catholic bishop who prefers the antipope over the real pope will pay taxes to him instead, and every bishop in the realm of the antipope controller will get the head of religion opinion bonus of 10 towards the antipope rather than the real pope.
The antipope will also have a claim upon the papacy, and by pushing this claim and winning the war, he will be installed as head of the catholic church, and the old pope will become an antipope with no successor. The split in the church will be gone upon the death of the ex-pope. The only real benefit to this is the +100 opinion modifier you'll get with the new pope, which will disappear upon succession.
While an antipope has the major benefits of tax and protection from excommunication, it will be a gradual drain on the church's moral authority. Each antipope in existence will decrease the MA of the church by 0.1% per year, which will over time reduce the rate of spread of Catholicism while increasing the spread of heresy. Due to this major disadvantage I would not recommend holding onto an antipope for any major period of time unless you deliberately want to become a heretic.