Have different Popes ever contradicted each other?

To add a supplemental answer to Jayarathina's excellent answer:

When a pope speaks in an infallible way, this is called speaking ex cathedra. This is not a ritualistic linguistic formula or a "label" like nihil obstat, it is simply a way to recognize those times when the pope gives a teaching that has been correct, is correct, and will remain correct forever. The key point is that the teaching is not "new", but expresses a known and unchanging truth on a matter of faith (what should be believed) and morals (what should and should not be done). Ecumenical councils are another organ of infallibility.

Now, to be clear:

• No pope has ever denounced an infallible teaching.

• Nor has one infallible statement ever been set against another one.

• Nor has any pope ever made a doctrinal error when speaking in this way:

  The broad fact, therefore, remains certain that no ex cathedra definition of any pope has ever been shown to be erroneous. (The Catholic Encyclopedia on Infallibility)

  The Catholic Encyclopedia link above discusses this further, for those interested in the details.

Note that ex cathedra is basically a clear statement on a particular topic, made by a particular person, and directed at a particular group of people. That's all. There have now been 266 popes over the course of 1981 years (32-2013). It does not take much to speak ex cathedra, and though it is not an everyday occurrence, many clear ex cathedra statements have been made. Given the enormous range of years, of different social attitudes, and of popes with different personalities and views, I personally think it is remarkable that Catholics do not have a huge pile of contradictory statements of this form. We should be very thankful.

Related:
• Can dogma ever be modified, and how does this relate to papal infallibility?
• Why did it take so long for papal infallibility to be defined?
• When does the Pope speak ex cathedra?

The part of the question concerned with a sitting pope revoking or contradicting an infallible statement made by a previous Pope requires us to define carefully just which statements unambiguously qualify as infallible statements, and which do not. Not every papal pronouncement was seen as infallible.

Michael Baigent says, in The Jesus Papers, page 13, that Pope Pius IX (1846-1878) was determined to be declared infallible, but knew that he would have to use guile to achieve this goal. Hence, the First Vatican Council was convened in late 1869. Baigent says its real aims were kept secret by a small group of powerful men that included three cardinals, all of whom were members of the Inquisition. No mention was made of papal infallibility in any of the documents circulated about the objectives and direction of the council.

John W. O’Malley, S.J. says, in A History of the Popes, pages 246-8, that as plans for the council developed, a broad agenda opened up - canon law, with the missions, with religious orders, and so forth, but the idea that the council should define that the pope was infallible in matters of faith began to be broached more and more often. Despite the anticipated broad agenda, the council dealt with only two items: the relationship between revealed truth and the powers of human reason, and the papacy, under the two headings of primacy and infallibility. Father O’Malley says that infallibility met with resistance from bishops for a variety of reasons, one of which was lack of clarity as to just what was meant by the term, and some simply did not believe it.

The pope was pronounced infallible on 18 July 1870. There had already been suggestions that in some circumstances the pope might be infallible, but it was the First Vatican Council that defined this as doctrine and defined the circumstances in which a statement by the pope should actually be regarded as infallible.

In 1854, Pius IX had defined the immaculate conception of the Blessed Virgin Mary, using the formula for ex cathedra statements that could be recognised as infallible, and this was recognised as the first use of this dogma of papal infallibility. Bishop Geofrey Robinson says, in Confronting Power and Sex in the Catholic Church, page 238, that Pope Pius XII declared infallibly on 1st November 1950, that Mary, the mother of Jesus, was assumed bodily into heaven, but Robinson regards this as a non-essential belief. Neither has been overturned, and regardless of Robinson's views on the ascension of Mary, neither of these statements is likely to be overturned by a pope in the foreseeable future.

Francis A. Sullivan SJ says in From Apostles to Bishops that Pope Leo XIII, in his bull Apostolicae curae of 1896, declared Anglican orders null and void. Although Leo did not pronounce this ex cathedra, officials of the Congregation for the Doctrine of the Faith have recently described Apostolicae curae as an infallible papal declaration. Any future negotiations for accommodation or unity with the Anglican Church could perhaps require a future pope to make a contrary pronouncement.

Robinson said (page 122) that some recent quasi-official opinions hold that when popes constantly repeat the same teaching, there is a point at which this repeated teaching becomes infallible. He says the Second Vatican Council set aside many things that had been consistently taught for more than a thousand years and would by this criterion have been infallible. Either repetition does not make a statement infallible, or the Second Vatican Council is an example of infallible statements being revoked.