Question: "...What exactly does papal infallibility mean and are there any limits to this?"
Fr. Brian Mullady, OP answers this in this HPR column. (See his second question.) Referencing "one of the outstanding theologians at Vatican I," Bishop Vincent Ferrer Gasser, he lays out what is essential to papal infallibility. (Vatican I is the council that defined papal infallibility.)
Who is the subject of the infallibility?
The whole Church is the subject of the infallibility in believing by an aid of the Holy Spirit, but acts in two specific organs to define what all believe:
[1] bishops with pope;
[2] the pope alone. ...The body cannot act apart from the head, though the head can act personally, and this includes the action of the body.
The Bishops may be gathered together in council, or in the case of ordinary Magisterium, they may be spread out in time and place.
The matter is definitive teaching on faith and morals:
...The word define must be taken in a strictly theological sense, as “to give a definitive judgment.” ..Such definitive teachings are by nature irreformable.
These irreformable teachings recognize truth divinely revealed in Scripture or in living Tradition, as well as truths which necessarily follow from revealed truths.
The form is the manifest intention to make a definition.
The pope must manifest his intention to make a judgment about doctrine.
Some wanted to limit the pope to a form [a particular verbal formula, as in the Sacraments], but ...“Already thousands and thousands of dogmatic judgments have gone forth from the Apostolic See; where is the law that prescribed the form to be observed in such judgments?” 5
In the case of ordinary Magisterium, the form is the repetition of a particular doctrine in various official Church documents.
The cause is the Holy Spirit working through the papal office.
The consent of the Church is a part of his infallibility, but not a condition de iure for the action of the Holy Spirit.
While some would like to limit the infallibility saying that the Pope is no longer infallible if he is in grave sin or if he privately embraces heresy, these conditions would in fact make void the promise of the Holy Spirit.
Many of the Fathers at Vatican I tried to introduce conditions into the definition of papal infallibility. The most famous ...was expressed: Thus the pope is infallible when he does what he can (papa faciens quod est in se est infallibilitas). This means that the pope must study the problem and consult with the bishops and with experts...
[The truth is to the contrary:] The pope cannot be bound to do what he can or to consult with others because these two conditions pertain to his conscience in the moral order, and is between him and God. ... A charismatic grace like the infallibility cannot depend on the pope’s conscience, which is a private affair, but on the pope’s public office within the universal Church. 4
Nor is it reasonable to say that the person of the Holy Father is no longer Pope if he should err and attempt to define a heresy. Since the Pope is indeed infallible while exercising his teaching office, he cannot err when defining doctrine. This is precisely the faithful's guarantee that the Pope cannot define any heresy.
...Infallibility [in the Church] is not absolute because this [absolute infallibility] is found in God alone. It is conditioned
1. by subject (the pope);
2. by object (a solemn definition of faith and morals); and
3. by cause (the action of the Holy Spirit).
Obviously, not every papal act falls under the infallibility [for example] if the pope makes a judgment on some practical matter, such as the morality of an individual war or use of capital punishment.
What a blessing papal infallibility is! Yes, sometimes definitive teaching is not as clear as we would like, but when we understand individual teachings in keeping with the whole body of Tradition, we can be assured of the truth of our faith. Thank you, Lord, for this amazing gift!