Infallibility

Vatican Council I defined what Catholics had always believed: that the pope, like the ecumenical (worldwide) councils, is infallible (preserved by God from error) when defining doctrine or morality for the whole Church. He is not personally infallible, but his office is.

God did not let us wonder and wander in darkness about the most important truths we had to know in order to fulfill our most important task in life, union with him. No human lover would allow that if he could help it. Neither did God. Papal infallibility, like every other Catholic dogma, is properly understood only by the primacy of love...

When the Church is infallible

1. "The Roman Pontiff, head of the college of bishops, enjoys this infallibility in virtue of his office, when as supreme pastor and teacher of all the faithful..., he proclaims in an absolute decision a doctrine pertaining to faith or morals" (LG 25).
2. "The infallibility promised to the Church is also present in the body of bishops, when together with Peter's successor, they exercise the supreme teaching office" (LG 25).
3. Even doctrines not explicitly labeled infallible can be binding on Catholic belief because "[d]ivine assistance is also given to the successors of the apostles, teaching in communion with the successor of Peter,...when, without arriving at an infallible definition and without pronouncing in a 'definitive manner,' they propose in the exercise of the ordinary Magisterium teaching ... of faith and morals. To this ordinary teaching the faithful 'are to adhere...with religious assent' (LG 25)" (CCC 892). Wise and good parents do not explicitly label everything they say to their children as "infallible", yet wise and good children trust them. Similarly, we should trust Holy Mother Church, the Church of the apostles, saints, and martyrs, the Church with a two-thousand-year-long-memory, much more than we trust our own opinions.
4. The sign the Church attaches to an infallible teaching is Christocentric: "When the Church through its supreme Magisterium proposes a doctrine 'for belief as being divinely revealed,' and as the teaching of Christ, the definitions 'must be adhered to with the obedience of faith'" (CCC 891).
   

Catholic Christianity:A Complete Catechism of Catholic Beliefs based on the Catechism of the Catholic Church by Peter Kreeft