A nation is a political entity associated with a people who share common cultural, ethnic, linguistic or ancestral ties. Nations are not always sovereign countries, nor are they always necessarily geographic in nature. A people can collectively consider themselves a nation without ever possessing self-government or territory. Quebec, for instance, is considered "a nation within a country," even though politically Quebec is a province of Canada. Various Indian tribes also call themselves "nation" though lacking sovereignty or their own land. Most commonly, however, nation is used as a synonym for a country or state.
In NationStates, nations are frequently founded and cease to exist in a regular cycle. Hundreds of nations are formed each day and hundreds more are deleted. To found a nation, a player need only specify a few small details (such as a unique name, flag, motto, currency and animal), then answer a short political survey before the desired nation comes into being. Other attributes such as population, economy, civil rights, WA membership and government category ("Inoffensive Centrist Democracy," "Father Knows Best State") change regularly based government decisions and other factors. Nations can exist for as short as 28 days or as long as the game lasts, depending on a player's longevity (and good behavior). Their involvement with the NS community can be limited purely to answering issues, voting on resolutions and posting on regional messageboards, or can branch out into roleplay on the forums, or political/military gameplay in R&D. It's all really up to the nation's owner.
NS nations can be either their own sovereign entities or territorial possessions of another. The sky is pretty much the limit as to what form a nation can take, whether it be a kingdom, a republic, a democracy or a hard-nosed dictatorship -- but since the effect of NS stats can be quite unpredictable at times, numerous players often roleplay their nations as something very much different from what the game categorizes them as (for example, a WA resolution changes a member nation to "Capitalizt," even though it has been a reliably communist state for years).
A nation-state is a self-governing nation (i.e., a people (nation) with their own government (state)), though just as with "nation," the term's definition is rather fluid.