What does papal infallibility actually mean?

Infallibility is one of the most misunderstood Catholic teachings, by Catholics as much as anyone.

What it is. Vatican I (1870) defined that when the pope speaks ex cathedra ("from the chair"), that is, when as shepherd of all Christians he defines a doctrine of faith or morals to be held by the whole Church, he possesses the infallibility Christ willed for His Church (CCC 891). The bishops in union with him, especially in an ecumenical council, share the same charism.

What it is not.

• Not impeccability. Popes sin; some have sinned spectacularly. Peter denied Christ.
• Not omniscience. Popes can be wrong about science, politics, prudential judgments, and even theology in their private opinions.
• Not inspiration. The pope receives no new revelation; he guards and clarifies what was given once for all (Jude 3; CCC 66-67).
• Not everything the pope says. Interviews, homilies, off-the-cuff remarks, even encyclicals are not automatically infallible, though they deserve religious respect (CCC 892).

How often has it been used? Explicitly, twice: the Immaculate Conception (1854) and the Assumption (1950). Many other teachings are infallible through the ordinary universal Magisterium, held always and everywhere.

Why it exists. Jesus promised that the gates of hell would not prevail (Mt 16:18), that the Spirit would guide the Church into all truth (Jn 16:13), and that whoever hears the apostles hears Him (Lk 10:16). Infallibility is a negative protection: God will not let His Church bind the faithful to error in essentials. It is a guarantee for us, not a privilege for him.