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The Catholic Church considers the Pope infallible, but only in limited circumstances. The Pope is a human being like the rest of us, and capable of sinning. However, when the Pope speaks ex cathedra, i.e. with papal authority, he is infallible.
How do Catholics know when the Pope speaks ex cathedra?
Vatican I defined that:
A true pope (not an antipope or heretic claiming to be the pope), when speaking on a matter of faith or morals (not economics or science) when speaking to the universal Church (not a letter to a friend) is guided by the Holy Spirit.
when the Roman pontiff speaks EX CATHEDRA,
that is, when,
- in the exercise of his office as shepherd and teacher of all Christians,
- in virtue of his supreme apostolic authority,
- he defines a doctrine concerning faith or morals to be held by the whole church,
The doctrine of papal infallibility was formally accepted at the First Vatican Council in 1870. This is not to say that it did not exist previously to that date: only that 1870 is when it became dogma - a teaching which all Catholics are required to believe. Exercises of infallibility are more difficult to identify when occurring prior to 1870. There have been some official rulings but there is no definitive list.
Since the Council, Popes have tried to be clear about precisely when they are speaking ex cathedra, with reference to the description in the Council text (the Dogmatic Constitution Pastor aeternus). This kind of statement is a Big Deal, because it is declaring something that Catholics must believe. The Pope and the rest of the Church leadership will want it to be widely understood that an infallible statement has been made - in the text of the declaration and by means of accompanying documents and public statements.
John Paul II said in an audience The Holy Spirit Assists the Roman Pontiff (24 March 1993):
The conciliar texts also indicate the conditions for the Roman Pontiff's exercise of the infallible Magisterium. They can be summarized in this way: the Pope must act as "the shepherd and teacher of all Christians," pronouncing on truths regarding "faith and morals," in terms clearly showing his intention to define a certain truth and to require definitive assent of all Christians.
The quoted phrases are from the Dogmatic Constitution: omnium Christianorum Pastoris et Doctoris, and de fide vel moribus. He gave the example of Munificentissimus Deus, in which Pius XII wrote (44-45):
For which reason, after we have poured forth prayers of supplication again and again to God, and have invoked the light of the Spirit of Truth, for the glory of Almighty God who has lavished his special affection upon the Virgin Mary, for the honor of her Son, the immortal King of the Ages and the Victor over sin and death, for the increase of the glory of that same august Mother, and for the joy and exultation of the entire Church; by the authority of our Lord Jesus Christ, of the Blessed Apostles Peter and Paul, and by our own authority, we pronounce, declare, and define it to be a divinely revealed dogma: that the Immaculate Mother of God, the ever Virgin Mary, having completed the course of her earthly life, was assumed body and soul into heavenly glory.
Hence if anyone, which God forbid, should dare willfully to deny or to call into doubt that which we have defined, let him know that he has fallen away completely from the divine and Catholic Faith.
In addition, John Paul noted
The conciliar texts also point out how serious is the Roman Pontiff's responsibility in exercising both his extraordinary and ordinary Magisterium. He thus feels the need, one could say even the duty, to explore the sensus ecclesiae before defining a truth of faith, in the clear awareness that his definition "expounds or defends the teaching of the Catholic faith" ... The Pope (who not only plays a role as head of the college of bishops in the definitions on faith and morals that the latter make, or as the notary of their thoughts, but also a more personal role both in the ordinary Magisterium and in his definitions) carries out his task by applying himself personally and encouraging study on the part of pastors, theologians, experts in different areas of doctrine, experts in pastoral care, spirituality, social life, etc.
This suggests that even when the Pope does speak infallibly, he doesn't do so out of the blue, but after wide consultation. This is another way in which news of an infallible statement could become commonly known, before it is even made.
Because there has been a valid Ex cathedra Papal decree that is clearly now false and which is no longer accepted by the Church we can regrettably no longer assume any Ex cathedra is infallible. The Ex cathedra statement by Pope Eugene the 4th in Cantate Domino 1441 is one such example. It proclaims and teaches for all Christians to believe that: if not subject to the Pope, even if you give your life to be burnt for Christ, you will go to hell: Pagans, Jews and heretics are listed. This proclamation is both valid as by Ex Cathedra definition and clearly false and clearly rejected effectively by current Church teaching especially Vatican Two. As catholics we are required to believe in the Vatical 1 infallible decree on Papal Ex cathedra Infallibility under pain of anathema as in excluded from the sacraments. Our Church also requires us to always follow our consciences once we have done our best to inform them. This means that one can be anathema for sincerely following ones conscience by believing that this Vat. 1 decree must logically be false. For thinking Catholics who love their Church this can be a very painful reality to come to terms with. Clearly the Church must come up with a response about some Ex cathedra statements such as the above or else humbly step down from its current claim regarding Ex Cathedra authority.
Papal Infallibility and when a Pope speaks Ex cathedra remain very unclear. Aside from canonization of saints, most experts will agree the Pope has spoken Ex cathedra at least twice:
Other than that opinions among experts vary widely. More can be found here: Wikipedia.
In 1870 Papal Infallibility was proclaimed an official teaching of the Church. However it was not declared an infallible dogma. But that is an issue of circularity. Furthermore, one of the two recognized invocations of the dogma of infallibility was made retroactively to cover the dogma of the Immaculate Conception of Mary, not to be confused with the Virgin Birth. There had been much debate and dissension among scholars ever since 1870.
Some clarification regarding the background of this troublesome doctrine may help.
The reason that the dogma of infallibility was invoked for the issues of the Immaculate Conception and the Assumption of Mary was that devotion to Mary was very strong among the faithful Catholics in the 19th century and had been since the Middle Ages. But there was no biblical basis for the popular beliefs about her sinless conception and her bodily assumption into heaven. Unlike doctrines about Jesus, for which there was always some scriptural basis, there was little mention of Mary in the Bible. Catholicism has always been a faith that sees the source of God’s Truth as both Scripture and Church Tradition, rather than Scripture alone, but the basis of Church teaching in the scriptures has always been very important. Another source of authority was therefore needed, hence the Infallibility of the Pope was needed to pronounce the two Marian dogmas.
One could also point out that the challenges to Church teaching from science, philosophy, and the growing numbers of atheist intellectuals, may have made the protection of the piety of the Catholic membership - concerning their beliefs in Mary - of great importance to the leadership of the Church at the time.
An interesting article on the political and sociological changes that affected the Church in the 19th century, and the rise of Marian piety, can be found here:
“The Catholic Church in Europe since the French Revolution”
Another article on the Catholic Church in the 19th century can be found here. Note: This article comes from a Catholic Journal: The Catholic Church and the Challenge of Modernity in 19th Century Europe.
Papal infallibility means that under certain conditions the Roman Pontiff (the Pope) cannot err when teaching about faith or morals. This is called speaking ex cathedra and its conditions include:
There is debate whether any ex cathedra statements were ever made. The Immaculate Conception and the Assumption have been cited as ex cathedra statements. Further, Canon Law 749 § 3 states:
No doctrine is understood as defined infallibly unless this is manifestly evident.
For this statement to be true, a doctrine must first be declared infallible before it meets the criteria of being manifestly infallible, which is circular logic.
There are already great answers to the question. I want to contribute with an analogy, which might help further clarify the concept of infallibility. It's related to the actual terminology used, ex cathedra.
The English word "cathedra" comes from the Latin cathedra. Among the meanings of the latter was that of the "chair of a professor", meaning that was also adopted in English. It is common in universities around the world (since the origin of universities, in medieval times) to hear of a given professor to hold the "chair of X", with X being a famous scholar. Sometimes the chair comes with the privilege of teaching a core lecture of the topic. We can think of the chair title as representing an authoritative position upon which the chair holder is to be trusted when lecturing about the topic related to the chair. Importantly, this authoritative position (i) emanates from the chair and not from the person holding it, (ii) it is official only when the chair is actually exercised (i.e. teaching), and (iii) pertain only to topics related to the chair. This is, anyone talking about the topic who does not hold the chair (e.g. a teaching assistant) does not have the same authoritative status. Similarly, the chair holder casually talking about the topic, say in the local bar, is not "sitting in the chair" and so cannot use the authority. And any other topic unrelated to the chair (say, weekend football) of which the chair holder talks is of course not authoritative.
The case of the Pope is like the above. Infallibility is a property of the "chair of St. Peter", not of the person of the Pope. It is exercised only through official channels (e.g. not when the Pope gives an interview to a newspaper), and must be related to topics upon which the chair has authoritative guidance, i.e. faith and morals.
