Which country or state is a point in?

Every country, state and county is stored as a shape, a polygon. To find which one holds your point, the computer checks which shape the point falls inside. In GIS this is called point in polygon.

Do it with country shapes, then state shapes, then county shapes, and you get the whole ladder: continent, country, region, district.

Many countries are made of several pieces, called a multipolygon. Hawaii and Alaska are part of the United States shape, even though they are far from the rest of it.

Some pieces sit inside another country. Llívia is a small Spanish town completely surrounded by France.

Some borders are disputed: Kashmir, Crimea and Western Sahara are well-known examples. Different maps draw them differently, so the answer depends on the boundary data the map uses.

A point exactly on a border line, such as a road down the middle of a frontier, can honestly be in both. Good tools say so rather than picking one.