In discussing the teaching authority — the magisterium — of the Church, most people put up resistance to the idea that any single man or collection of men can be held infallible. This even holds true for many Catholics, whether their dissent is religiously conservative (SSPX) or liberal (Catholics For Choice).
But as we remember from “The sola scriptura problems”, Jesus promised the guidance of the Holy Spirit to the Church (Jn 14:26, 16:13). If we hold the Holy Spirit — God — to be trustworthy and reliable (Rom 3:3-4; 2 Tim 2:13), then that guidance must necessarily impart infallibility to the Church’s teachings. And, indeed, St. Paul calls the Church “the pillar and foundation of truth” (1 Tim 3:15). Nor can the Church lose that guidance without breaking Christ’s other promise: “And surely, I am with you always, to the very end of the age” (Mt 28:10 NIV).
Let’s consider it from another angle: Infallibility, in doctrinal terms, functions much like the American judicial principle of stare decisis (“let the decision stand”). Under stare decisis, lower courts aren’t free to controvert decisions of upper courts, especially not those of the Supreme Court. The only difference is, while SCOTUS holds itself able to overrule previous decisions at its own level, infallibility goes forward in time to bind future councils and popes: Pope Benedict could no more dispense with the doctrine of the Immaculate Conception than he could overrule the doctrine of the Trinity.
This starts to give us an answer to those who fear the power of the Keys of Peter as a “blank check”. Theologian and biographer George Weigel tells this story:
During the Second Vatican Council, Pope Paul VI suggested that the Council’s basic text, the Dogmatic Constitution on the Church, include the statement that the Roman Pontiff “is accountable to the Lord alone.” The Council’s Theological Commission told Pope Paul politely but firmly that that simply wasn’t the case. Any pope, the commission pointed out, is also accountable to God’s revelation, to the fundamental structure of the Church given it by Christ, to the seven sacraments, to the creeds, to the doctrinal definitions of earlier ecumenical councils, and to “other obligations too numerous to mention,” as the commissioners delicately put it.
To further understand the limits of infallibility, let’s look at the First Dogmatic Constitution on the Church of Christ, the product of the fourth session of Vatican I which dogmatically defines papal infallibility. The document states three conditions that must be met for infallibility to be presumed:
1. The pope is speaking “in the exercise of his office as shepherd and teacher of all Christians”;
2. He is speaking “in virtue of his supreme apostolic authority”;
3. He is “defin[ing] a doctrine concerning faith or morals to be held by the whole church”.
These three conditions define what it means for the Pope to speak “ex cathedra”. Not every tentative opinion or flip comment is sheltered; the Pope’s bets on soccer games aren’t divinely covered. It also doesn’t cover matters of discipline or governance; a later pope is free to change those as circumstances warrant and his judgment thinks best.
More importantly, the Pope isn’t guaranteed infallibility on scientific matters, except insofar as they impinge on matters of faith and morals. For example, he can’t either affirm or deny evolution as a scientific fact; he can, however, dispute authoritatively whether evolution “proves” the non-existence of God (another discussion entirely, so no rabbit holes here!). We may have a quibble here on the social sciences, since they purport to study the things humans do and the Church's magisterium concerns the things humans ought to do ... but that's an issue for another time.
Similarly, this special protection extends to “the body of bishops when, together with Peter’s successor, they exercise the supreme Magisterium” in an ecumenical council. (“Truly, I say to you, whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven” [Mt 18:18].) Like the infallibility of the popes, it extends forward over time, especially when an opposing statement is declared anathema (Gr. “accursed”).
There is also the “exercise of the ordinary Magisterium”, which depends less on the dogmatic definitions of Pope or Council and more on the consistent teaching of the doctrine over the life of the Church. For instance, Bl. John Paul II’s apostolic letter Ordinatio Sacerdotalis, which declared that “the Church has no authority whatsoever to confer priestly ordination on women”, is held to be infallible through the ordinary magisterium rather than through the specific power of the keys.
There is a slight difference: whereas dogmatic definitions are to be held with “the assent of faith”, the ordinary exercise is held with “religious assent”. The practical effect, though, is still the same: the teaching is irreformable, and opposition or denial constitutes a breaking of communion.
There are teachings of the Church which aren’t covered by either the ordinary universal magisterium or the extraordinary exercises, some of which Catholics can in good conscience disagree with one another. However, the safe bet for the ordinary untrained layperson — or even for the trained theologian — is to stick to the Catechism, and to research and think about those teachings that “stick in the craw” until one understands.
The key takeaway, though is this: Infallibility asks us to trust not just the teachings of the Church but in the Holy Spirit’s guidance of the Church. It has no bearing on the character or intelligence of the bishops; Alexander VI was as venal and corrupt a gnome as ever wore the Shoes of the Fisherman, yet he taught no error! And it’s not as if the teachings come from nowhere; even doctrines like the Immaculate Conception and the Assumption arise from years — even centuries — of discussion and analysis.
Once we understand this, we’re in a better frame of mind to tackle the issue of apostolic succession.
Update: May 2, 2011In view of the discussion in the combox, I'd like to add some clarification on the status of the Society of Saint Pius X (SSPX):
On June 20, 1988, Archbishop Marcel Lefebvre and retired Bishop Antônio de Castro Mayer, acting against Canons 1013 and 1382 of the Code of Canon Law, illicitly consecrated four bishops without mandate from the Holy See: Bernard Fellay, Bernard Tissier de Mallerais, Richard Williamson (of later Holocaust-denying notoriety) and Alfonso de Galarreta. A couple of weeks later, Pope John Paul II issued a motu proprio declaring that the consecration ceremony "constitutes a schismatic act and that the participants — not naming +Castro — "incurred the grave penalty of excommunication envisaged by canon law" (Ecclesia Dei, 3).
Moreover, in "an appeal both solemn and heartfelt", he reminded everyone linked to the SSPX that "formal adherence to the schism is a grave offense against God and carries the penalty of excommunication decreed by the Church's law." According to Canon 1364, schism carries with it a latae sententiae excommunication (i.e., the guilty has earned the penalty by the fact itself, not by judicial sentence — effectively a mandatory punishment).
Since then, there have been two notable events which slightly modify our understanding. The priests of the order are validly (if not licitly) ordained; the four bishops, similarly, were validly but not licitly ordained. The pontifical commission erected by Ecclesia Dei and named after it declared that attendance of a Mass celebrated by an SSPX priest wasn't licit unless one is physically or morally impeded from attending a Mass celebrated by a priest in good standing. Canons 1322 and 1323 also modify the formal judgment of what constitutes "adherence".
Unfortunately, the former president of Ecclesia Dei, Cardinal Darío Castrillón Hoyos, made a series of statements in a number of interviews that merely confuse the matter, especially his statement that the 1988 consecrations gave rise to a situation of separation, even if not a formal schism. While Ecclesia Dei, since it treats matters of discipline, doesn't fall under the protection of infallibility, it is a papal decree and therefore must be treated as having legal force. Moreover, when B16 lifted the excommunications of Bps. Fellay, de Mallerais, Williamson and de Galarreta, he confirmed that SSPX has no canonical standing and its ministers don't exercise a legitimate ministry.
For these reasons, we should conclude that the SSPX is still de facto and de jure in schism. I can admit to error in attributing the source of their schism, which lies not in holding a doctrine to be in error (but see the article by Sandro Magister linked in the combox) but in disobeying the magisterium of the Church. Again, though, it comes down to whether one trusts the guidance of the Church by the Holy Spirit.
 The Society of St. Pius X (SSPX) is a schismatic traditionalist organization that holds the teachings of Vatican II — and the popes who support it — to be in error. Catholics For Choice oppose the Church’s teachings against abortion.
 Weigel, The Courage to Be Catholic (2002), p. 117-118.
 Op. cit., 4:9.
 “A distinguished Catholic philosopher who thinks himself extremely orthodox once said, ‘If the Pope said “2 + 2 = 5”, I’d believe him.’ Another distinguished philosopher, just as committed to the papacy as his colleague, gave the correct and far more orthodox response: ‘If the Pope said, “2 + 2 = 5”, I’d say publicly, “Perhaps I have misunderstood His Holiness’ meaning.” Privately, I would pray for his sanity’” (Weigel , p. 118).