An Islamic equivalent to the Pope would be the Caliph, the leader of Islam on earth.

The Papacy is based on Christ's direction to Peter "You are the rock upon which I will build my church" Peter was the first Pope, the leader of the church on earth. The Catholic church has maintained a pope (or leader) ever since. Catholics believe that all Popes are the successors to Peter, not the previous pope.

The Pope makes the rules for the church, which has at times throughout history caused difficulties for the church, particularly during the middle ages. There's a lot more to this subject that would take pages to write, I would suggest wikkipedia or how stuff works.

http://people.howstuffworks.com/papacy.htm