Who runs the Catholic Church?
Hierarchy of the Catholic Church
The pope is the bishop of Rome. He is also, by virtue of that office:
Catholicity
The Catholic Church's doctrine of infallibility derives from the belief that the authority Jesus gave Peter as head of the church on earth has been passed on to his successors, the popes. Relevant Bible verses include ; "And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it. I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven."
The Latin and Eastern Catholic Churches together form the "Catholic Church", or "Roman Catholic Church", the world's second largest religious denomination after Sunni Islam, and the largest Christian denomination, as well as its largest Catholic church, comprising over half of all Christians (1.27 billion Christians of 2.1 billion) and nearly one-sixth of the world's population. Richard McBrien would put the proportion even higher, extending it to those who are in communion with the Bishop of Rome only in "degrees". It comprises 24 component "particular Churches" (also called "rites" in the Second Vatican Council's Decree on the Eastern Catholic Churches and in the Code of Canon Law), all of which acknowledge a primacy of jurisdiction of the Bishop of Rome and are in full communion with the Holy See and each other.