This is the belief that the Pope, when defining a teaching on faith or morals that is to be held by the whole church is able to make such pronouncements without error.
First defined as a Dogma, or required belief by the First Vatican council in 1870, Papal Infallibility was used in 1950 to dogmatically define the Assumption of the Virgin Mary i.e. the view that she was taken directly into heaven at the end of her life.
Teachings made by the Pope in this manner are known as ex cathedra (Latin: ‘from the chair’). There is no definite list of infallible teachings, but apart from the 1950 defintion, the 1854 dogma of the Immaculate conception of the Virgin Mary is also considered infallible.
The teaching does not imply that the Pope is without sin (known as impeccability as opposed to infallibility) or unable to make errors of judgment, but acting with the authority as successor to Peter (Matthew 16:18), he is able, through the aid of the Holy Spirit, to pronounce infallibly. In addition, ex cathedra teachings are irrevocable and not subject to the consent of any other authority within the church.
The definition given at Vatican I is as follows:
….”we teach and define as a divinely revealed dogma that 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, he possesses, by the divine assistance promised to him in blessed Peter, that infallibility which the divine Redeemer willed his Church to enjoy in defining doctrine concerning faith or morals. Therefore, such definitions of the Roman Pontiff are of themselves, and not by the consent of the Church, irreformable.”
Finally, it should be noted that infallible teachings can also be made by ecumenical church councils (such as Vatican I) and not just directly by the Pope himself.