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).

In discussing the possibly of the Church's ordaining of women to the priesthood, I generally like to note that the Church's teaching is infallible, and that it cannot be changed. For me as a Catholic, this means that the teaching is decisive, authoritative, binding (both to my intellect and my will), and final. One tactic which I have noticed increasingly is the insistence that the teaching is not infallible--this even by some Catholics who presumably do accept that the Church can teaching infallibly on matters of doctrine and morality. Their argument hinges largely on an interpretation from Canon Law (itself fallible) which states that in order for a teaching to be infallible, there can be no speculation as to whether or not the doctrine has been taught infallibly.

Such speculation exists in the case of the teaching regarding priestly ordination. Therefore, it is concluded by some that the teaching against ordaining women to the priesthood is not infallible. This speculation is apparently not dispelled by Pope John Paull II's Ordinatio Sacerdotalis, which stated rather conclusively that

Priestly ordination, which hands on the office entrusted by Christ to his Apostles of teaching, sanctifying and governing the faithful, has in the Catholic Church from the beginning always been reserved to men alone. This tradition has also been faithfully maintained by the Oriental Churches....Although the teaching that priestly ordination is to be reserved to men alone has been preserved by the constant and universal Tradition of the Church and firmly taught by the Magisterium in its more recent documents, at the present time in some places it is nonetheless considered still open to debate, or the Church's judgment that women are not to be admitted to ordination is considered to have a merely disciplinary force.

Wherefore, in order that all doubt may be removed regarding a matter of great importance, a matter which pertains to the Church's divine constitution itself, in virtue of my ministry of confirming the brethren (cf. Lk 22:32) I declare that the Church has no authority whatsoever to confer priestly ordination on women and that this judgment is to be definitively held by all the Church's faithful.