When is a Pope infallible?

The pope is infallible when he speaks "ex cathedra," or from the chair in his official capacity as pastor and teacher. The matter must be a revealed truth, on matters of faith or morals. Revealed meaning guided by the Holy Spirit. He must intend to teach infallibly and make this known at the time of that teaching.

In 1870, sixteen years before Vatican I, the Immaculate Conception was defined as an infallible doctrine. The last pope to release a second document as infallible was in 1950 when Pope Pius XII solemnly defined the belief in Mary's Assumption is part of the faith. That belief had been an accepted part of religious belief before that time. Therefore, we must understand that the popes do not release what is considered infallible teachings on a regular basis. The pope is never considered infallible in his personal or private views.

Papal authority was given to Peter when Jesus and he had the dialogue in Matthew 16:13-19, often called "The Confession of Peter":

During his life, Peter was never officially declared bishop of Rome or the pope. However, because of his work and his role as head of the Church, he is recognized at the first pope.

This papal authority as the Bishop of Rome (later called Pontifex, Pontiff or "Bridge") passes from Peter to his successors. Therefore, every pope since Peter is considered the immediate successor of Peter, and not of that pope's immediate predecessor.
The Roman Pontiff, the Successor of Blessed Peter in primacy, has not only the primacy of honor, but also supreme and plenary power of jurisdiction throughout the universal Church, both in matters which pertain to faith and morals, but also in those which have to do with the discipline and order of the Church. (Canon 218, Paragraph 1)