Jesus said to him in reply, "Blessed are you, Simon son of Jonah. For flesh and blood has not revealed this to you, but my heavenly Father. And so I say to you, you are Peter, and upon this rock I will build my church, and the gates of the netherworld shall not prevail against it. I will give you the keys to the kingdom of heaven. Whatever you bind on earth shall be bound in heaven; and whatever you loose on earth shall be loosed in heaven."
This quote is interpreted by Catholics as Jesus making Simon the religious leader of Christianity, and the Pope is simply in a long line of successive "shepards of the flock" that started with Simon.
It can be interpreted in other ways, obviously, but this quote is the foundation for the Papacy.