An "ex cathedra" statement is an exercise of the Pope's extraordinary magisterium, or teaching authority. The term "ex cathedra" is Latin for "from the Chair" — as in Chair of Peter, one of the symbols of a bishop's power as successor of the Apostles. The term "cathedral" — which serves as the seat of the bishop in a diocese — also comes from the word "cathedra."
For a statement to be "ex cathedra," it must meet several criteria as put forward by "Pastor Aeternus," or the "First Dogmatic Constitution on the Church of Christ" promulgated at the First Vatican Council:
[W]e 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.
Two ex cathedra proclamations have been issued by the Church: once in 1854 and once in 1950. Both times had to do with theology relating to our Blessed Mother. In 1854, Pius IX infallibly declared the dogma of the Immaculate Conception, and 96 years later in 1950, Pius XII declared and defined the dogma of the Assumption of Mary.
Watch the full episode: "The Download—Papal Infallibility."