From the Catechism of the Catholic Church:

889 In order to preserve the Church in the purity of the faith handed on by the apostles, Christ who is the Truth willed to confer on her a share in his own infallibility. By a “supernatural sense of faith” the People of God, under the guidance of the Church’s living Magisterium, “unfailingly adheres to this faith.”

890 The mission of the Magisterium is linked to the definitive nature of the covenant established by God with his people in Christ. It is this Magisterium’s task to preserve God’s people from deviations and defections and to guarantee them the objective possibility of professing the true faith without error. Thus, the pastoral duty of the Magisterium is aimed at seeing to it that the People of God abides in the truth that liberates. To fulfill this service, Christ endowed the Church’s shepherds with the charism of infallibility in matters of faith and morals. The exercise of this charism takes several forms:

891 “The Roman Pontiff, head of the college of bishops, enjoys this infallibility in virtue of his office, when, as supreme pastor and teacher of all the faithful – who confirms his brethren in the faith he proclaims by a definitive act a doctrine pertaining to faith or morals. . . . The infallibility promised to the Church is also present in the body of bishops when, together with Peter’s successor, they exercise the supreme Magisterium,” above all in an Ecumenical Council. When the Church through its supreme Magisterium proposes a doctrine “for belief as being divinely revealed,” and as the teaching of Christ, the definitions “must be adhered to with the obedience of faith.” This infallibility extends as far as the deposit of divine Revelation itself.

892 Divine assistance is also given to the successors of the apostles, teaching in communion with the successor of Peter, and, in a particular way, to the bishop of Rome, pastor of the whole Church, when, without arriving at an infallible definition and without pronouncing in a “definitive manner,” they propose in the exercise of the ordinary Magisterium a teaching that leads to better understanding of Revelation in matters of faith and morals. To this ordinary teaching the faithful “are to adhere to it with religious assent” which, though distinct from the assent of faith, is nonetheless an extension of it.

From a comment on a blog discussing Anglicanorum coetibus:

“Papal Infallibility for me is the dogma that prevents me from joining the Roman Catholic Church.”

In my work with potential converts to the Catholic faith, I can’t remember how many times someone has told me, “I just can’t accept ______” (fill in the blank).  It might be the Immaculate Conception, the Assumption of the Blessed Virgin Mary, papal infallibility, transubstantiation, or some other Catholic doctrine.  My response is always, “Tell me what you think the Church teaches about that.”  When they tell me, it’s not surprising they’re having trouble accepting it.  I wouldn’t be able to, either.

Taking just the matter of papal infallibility, it’s remarkable how many people confuse it with impeccability, thinking we’re claiming that the Pope can’t sin.  Or they imagine that it applies to every single thing he says, and every random thought that crosses his mind.  Frequently, people have a too-narrow idea of it, not understanding that it belongs to the whole Church, and flows from an adherence to the Magisterium.

Some people think it gives a kind of super-power to the Pope, when in reality it limits him simply to teaching the truth.

So if someone says they can’t accept infallibility, or the Marian dogmas, or any other aspect of the Faith, let’s make sure what they “can’t accept” is what the Church really teaches.