Strictly speaking, only persons may be infallible, as only persons may be fallible. The term "infallible" simply means immunity from error, and it would be a strange sort of category mistake to describe an assertion that is true as "infallible". Assertions as such are neither "fallible" nor "infallible", they are either true or false. If an assertion is true, we may call it "authoritative" in the sense that the truth demands assent, but we do not say that it is "infallible" because it is either pointless or trivial to say that truth is "immune" to falsity. Hence, very strictly speaking, we do not say that the teachings of the Ordinary Magisterium are "infallible", we say that they are "authoritative". In a looser way, of course, we do speak of the "indefectibility" of the Church, and by this we usually tend to mean that we believe the Church to be immune to error, but this is a sort of analogous statement by which we reify an institution that would not ordinarily be capable, qua institution, of infallibility tout court.
The Pope, by contrast, is infallible, because he is a person. His teachings are authoritative because of the authority of the office he holds, as I maintained in my earlier posts in this thread, by virtue of a very positive influx of divine grace. As the First Vatican Council formulated it:
The Roman Pontiff when he speaks ex cathedra,that is, when acting in the office of shepherd and teacher of all Christians, he defines, by virtue of his supreme apostolic authority, doctrine concerning faith and morals to be held by the universal Church, possesses through the divine assistance promised to him in the person of St. Peter, the infallibility with which the divine Redeemer willed his Church to be endowed in defining doctrine concering faith or morals; and that such definitions are therefore irreformable of themselves, and not from the consent of the ChurchThat last bit needs a little explication. It is important to note that the Church's belief in the infallibility of the Pope is not itself a Papal teaching, that is, there is no circularity involved in claiming this charism for the Pope: in addition to having been an element of the consensus fidelium in one form or another for quite some time, it is here defined by an Ecumenical Council, and the authoritativeness of Ecumenical Councils is not questioned by orthodox Christians (though, obviously, certain minority communities within the Christian community do question whether certain particular councils were genuinely "ecumenical" in the authoritative sense). There are, I believe, some folks who call themselves "Christians" but who reject the authority of Ecumenical Councils to define the Christian faith, but as I regard that point of view as utterly incoherent I will leave to others the task of trying to enter into dialog with such persons.
Another point that bears emphasizing is that infallibility differs from revelation in the following sense: revelation is always of new truth, while infallibility refers to the charism of preserving what has already been revealed. That is, when the Pope enjoys the charism of infallibility it is not the case that he is thereby enjoying a charism for the teaching of new, revealed truths coming down from heaven straight into his mouth. He is merely directing the Church's interpretation of what is already known via revelation. In this sense, the Pope's charism of infallibility is an instrument of the development of doctrine as Newman understood it. (It is interesting to note that Newman was not, at first, particularly comfortable with the Council's definition, because he believed that the consequence of defining the charism would, in fact, be to provide fodder for calling into question what ought to have been obvious about the Pope's teaching office.)
Another way to put this point would be to note that infallibility is not inspiration, that is, it is not, in fact, the voice of God speaking in the Pope's heart. It is, rather, a providential aid that the Pope enjoys by divine grace. Hence it is the Pope himself, and not God, who is the author of the teachings that are made by virtue of the charism of infallibility. Again, the charism of infallibility is only enjoyed by the Pope under certain very restrictive conditions, and one must keep those conditions in mind, along with the fact that infallibility is not at all the same thing as impeccability, that is, immunity from sin. No Pope enjoys impeccability as a charism of his office. According to the Ordinary Magisterium, Our Lord and Our Lady are the only human beings to enjoy impeccability (though it has sometimes been maintained that the Apostles, after Pentecost, were aided by divine grace in avoiding sin).
The standard schema, then, is often put this way: by revelation, God speaks his divine word; by inspiration he projects it into the human community; by infallibility he protects it from corruption within that community. Standard schemata, however, are made to be challenged, so it comes as no surprise to find that this one is challenged, too. The first element is challenged by the atheist or other non-Christian who simply does not accept the fact that knowledge of God is possible, or that it is possible in the way that Christians maintain that it is. The second is challenged by theists outside of the Judaeo-Christian tradition, who do not accept the word as preached by the prophets and teachers within the Tradition. The third element is challenged by a rather motley crew that includes probably a little bit of everyone who is not Catholic. For the Catholic, however, there is no way around this one, given the ecclesiology behind the teaching: if the Church herself, through the Ordinary Magisterium, teaches that the Pope is infallible under the circumstances described above, there is no logically consistent way to deny the teaching without simultaneously denying the authority of the Church to teach what is to be believed de fide.
This raises the very important issue of the consensus fidelium, that is, the nature of infallibility as predicated, not of any one particular individual, but of the body of Christian faithful, that is, the Church qua collective body of individual believers, to whom God has entrusted the fullness of his revelation. The Second Vatican Council identified this form of infallibility as the precursor to that enjoyed by the Pope and the College of Bishops (see, for example, Lumen Gentium, 2.12). An interesting facet of this form of infallibility is rather easier to see if one imagines the situation of the neophyte in the second and third quarters of the first century, before any of the New Testament texts had been committed to writing. During the period immediately following the death of Our Lord there was a variety of groups claiming to be followers of the Christ, but they certainly did not all agree on the nature of what it meant to be such a follower. Some versions of the story were decidedly heretical. These versions came later to be committed to writing as well as the more familiar (to us) orthodox texts, but for the neophyte of the years A.D. 33-80 there were no Gospel accounts other than those that were preached orally in the local house-churches in which the Christians met. The neophyte had to trust that the word he was hearing was authentic and authoritative. The only way to have such trust was to accept, by virtue of divine grace, the truth of what was preached. Certainly the Holy Spirit would not provide divine grace to believe a false account: it is an article of our faith that orthodoxy became what it is by virtue of the inspiration of the Holy Spirit himself. Of course, from a secular point of view, orthodoxy triumphed over heresy simply by strength of numbers, and then the victorious party suppressed the vanquished, as victorious parties usually do. There is no reason for the believer to accept the secular version of the story, however, since it is underdetermined by the data, but that fact is actually beside the point here, since I am addressing not the atheist or the non-Christian, but the person who challenges the third element of the schema, infallibility.
So it is the teaching authority of the Christian community itself that demands the assent of the neophyte on the grounds that said community is under the inspiration of the Holy Spirit. It is not strength of numbers, obviously, that makes the community infallible, as though the truth could be determined by majority vote, but rather this inner spiritual unity emanating from the Holy Spirit and directing them in the truth. It is particularly important to note that it was under the inspiration of the Holy Spirit that the Christian community produced the texts that now form the New Testament: the authority of these texts derives strictly from the authority of the Church to promulgate them as authoritative. In this context it is worth pointing out that the expression consensus fidelium, along with the others that are often used in the same way (universitas fidelium, universalem consensum), does not imply that there is now, or that there ever was, a time in which all Christians believed precisely the same thing in the sense that they all assent to precisely the same set of assertions. The consensus refers to that which unites Christians, namely, their belief in what is true and that they are inspired to believe by the Holy Spirit. Anyone who denies that the Church has this inspiration has no reason to believe in orthodoxy rather than heresy. That is, if you deny that the Church enjoys this charism, then there is no reason for you to be a Christian rather than a Gnostic, or an Arian, or indeed a Jew or a Muslim or even an atheist, because there is no reason for you to accept anything that the Church has ever taught, including such things as the divinity or resurrection of Our Lord or any of the other familiar elements of the Gospel. The charism to teach authoritatively, in other words, is centrally essential to all Christian belief. Indeed, without it there is no Christian belief.
Most non-Catholics, of course, do not deny this, just so long as the word "Church" is taken to mean "body of believers". There seems to be some hesitation about associating the term with any particular institutional body of believers. It is here, however, that the office of the Successor of Peter becomes essential to the Catholic position. As early as the Council of Ephesus and the dispute with Nestorius there is evidence that it is the unity of the College of Bishops with the Successor to Peter that gives an Ecumenical Council its magisterial authority. It was this aspect of the Petrine office that was the guarantor of the Council's authenticity. In the person of the Pope we find a sacramental image of the infallibility of the entire Christian community--a manifestation of that unity in the Spirit that is the special charism of the Church, but that unity is made explicit by residing in a special way in a single individual, the Pope. His infallibility belongs to him not as a private theologian or Christian person, but as a public person, the single head of the universal Church. Accordingly, the individual man who happens to fill the office of Pope enjoys infallibility only when he acts qua Pope, and as I have pointed out in earlier posts, he does not act qua Pope when he teaches error, he can only act qua Pope when he teaches what is true. But when he does, in fact, teach what is true, it is the belief of the Church that he teaches the truth by virtue of a special charism of the Spirit that protects him from error.
Although the Pope enjoys a special charism in this respect, he does not enjoy it in isolation from the rest of the Church. It is qua head of the Church that he enjoys the charism, after all. Hence the Pope is expected to take the universal Church into consideration when asked to settle a matter affecting the whole Church: regional and ecumenical councils, episcopal synods, the views of the College of Cardinals, the views of philosophers and theologians, all must be taken into account in the process of settling an issue. It is simply not the case that the Pope, qua Pope, thinks through an issue on his own and dictates by fiat what is to be believed de fide.
The two most famous instances of a Pope explicitly exercising his charism of infallibility are the definitions of the dogmata of the Assumption of Our Lady and of the Immaculate Conception of Our Lady. While these are the most famous instances, however, it is by no means clear that they are the only instances, in spite of what is often taught in RCIA classes. It has sometimes been assumed that, in order for a Papal decree to count as taught in accordance with the charism of infallibility, the Pope must explicitly claim, in the text of his decree, that he intends to exercise his charism, by using words along the lines of "in communion with the whole Church and in my capacity as the Successor of Peter, I solemnly declare and define etc.", but the Counciliar definition sets no necessary and sufficient conditions of this kind for infallible pronouncements. All that is required is that the Pope intend to exercise the charism; whether he makes explicit that he is doing so is up to him. Granted, from an epistemological point of view, such formulas as the one above are useful, because they help to avoid ambiguity on the point. However, I think that it is fair to say that, for example, John Paul's Ordinatio sacerdotalis is an instance of the charism of infallibility in spite of the fact that it does not contain the same sort of declarative wording that the dogmata of the Assumption and Immacualte Conception contain. But Ordinatio sacerdotalis has become a rather controversial document, and perhaps a clearer declarative statement would have been in order. Be that as it may, suffice it to repeat that the Council did not require any specific verbal formula.