State pretty much always used to mean a high governing authority for an area. Nation is similar, but implies a certain level of social cohesiveness. Here's what wikipedia has to say about the difference:
The state is a political and geopolitical entity; the nation is a cultural and/or ethnic entity. The term "nation-state" implies that the two geographically coincide, and this distinguishes the nation-state from the other types of state, which historically preceded it.
The United States of America pretty much spoiled "state" it in someits original sense, for purely historical reasons. Basically the individual 13 colonies that made it up wanted to act in concert, but keep all governing authority over their own citizens to themselves. Hence "United States".
After a few years this proved a bit unworkable (13 states each with their own currency and charging each other import and export tariffs kinda sucked for trade), so they got together and created single over-government to cover the entire federation of 13 states (a "federal government"). They still tried to leave a lot of power up to the individual states though. Over the years a lot of that power has bled up to the Federal government, so technically the USA is really one state. However, the name USA had stuck by then, and we still call our second-tier governing units "states".