The papacy is unusual among religious offices in that its powers are written down and are legally operative.
Under Catholic canon law the pope has full, supreme, immediate and universal ordinary power over the whole church, which he can always freely exercise. In practice: he appoints bishops, and can remove them; he is the final court of appeal, and no appeal lies from his judgement; he can legislate for the entire church by his own act; he convokes ecumenical councils, presides over them and confirms their decrees, without which they are not binding; and he can define doctrine.
The definition of 1870 set out two things that are commonly run together and are distinct. The first is universal jurisdiction, which is the governing power just described and is exercised daily. The second is infallibility, which applies only when the pope speaks as universal pastor to define a doctrine of faith or morals for the whole church, and which has been invoked explicitly on perhaps one occasion since, in 1950. Almost nothing a pope says falls under it, including most encyclicals and everything said in interviews.
Alongside all this runs something no other religious office has: the Holy See is a sovereign subject of international law. It maintains diplomatic relations with most states, signs treaties, and holds a permanent observer seat at the United Nations. This is separate from Vatican City, which is a territory created in 1929 to give the See a physical base after the Papal States were absorbed into Italy.
It is worth noticing when the jurisdictional claim reached its maximum. It was defined in 1870, in the same decade in which the papacy lost its temporal territory entirely. The strongest statement of spiritual authority arrived as political authority collapsed, and historians of the period do not treat that as coincidence.