In BE, the word *state* usually refers to an area controlled by one government.
The word *nation* can mean the same thing, but can also mean a group of people linked by culture, who for various historical reasons, live in several different states. An example is "the Kurdish nation".

Within the UK, in common usage, the word *state* is sometimes used to mean something controlled and funded by central government, for example "state school" as opposed to "private school".