What does it mean to say that the pope is infallible? Does this mean that he never makes any mistakes, never sins, or that he is perfect?

The doctrine of Papal Infallibility [1] was formally defined at the first Vatican council in 1870, but its roots run much deeper:

When Jesus went into the region of Caesarea Philippi he asked his disciples, "Who do people say that the Son of Man is?" They replied, "Some say John the Baptist, others Elijah, still others Jeremiah or one of the prophets." He said to them, "But who do you say that I am?" Simon Peter said in reply, "You are the Messiah, the Son of the living God." 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." (Matthew 16:14-19).

We can trace in an unbroken line all of the popes, from Benedict XVI back to Peter the apostle; and just as we believe that the bishops are the successors of the apostles, we believe that the Pope, as earthly head of the Church, is the successor to the "head" apostle, Saint Peter. As such, he is the visible head of the Church on earth and the vicar of Christ, who Is the Head of the Church (Ephesians 5:23).