The Vatican Council of 1870 defined two things about the bishop of Rome, and they are frequently run together. The first is primacy of jurisdiction: that the pope has immediate and ordinary authority over every church and every believer, not merely a right of appeal or a first place among equals. The second is infallibility, and the council fenced it in tightly. The pope teaches infallibly only when he speaks ex cathedra, from the chair of Peter, acting in his office as pastor and teacher of all Christians, and defining a doctrine concerning faith or morals to be held by the whole church.
Everything outside that description is untouched. Papal opinions on politics, science, prudential judgement and church discipline are not covered. Encyclicals as such are not covered, though they may restate what is already held infallibly. The pope's personal holiness is emphatically not covered, a point Catholic apologists make with reference to the tenth-century papacy, which was disreputable by any standard.
The practical result is a very short list. The definition of the Assumption of Mary by Pius XII in 1950 is the one exercise universally agreed to satisfy the conditions since 1870. The Immaculate Conception, defined by Pius IX in 1854, is regarded as an exercise of the same charism before it was formally described. Some theologians count a small number of other statements; there is no official register, which is itself a source of argument.
Catholic theology also holds that the ordinary and universal teaching of the world's bishops in union with the pope is infallible, which covers far more ground than papal definitions do. Infallibility, on this account, is a property of the church's teaching that the papal office exercises in a particular mode rather than a personal privilege of one man.