A nation is an ethnic construct. It describes a people with a common heritage, common language and customs, a shared history. A state is a political construct. A state has defined borders, a population to rule, a government that controls it, sovereignty over its own affairs (in other words, the government isn't held hostage by an occupying army), and an ability to carry out foreign policy. A state, then, is an independent country. A nation is, broadly, a distinct ethnic group.
If a state is populated by one nation, it is a nation-state. Portugal is a good example, Iceland is another, Japan is a third. An overwhelming majority of the population identifies itself with one nation, and so the state has a unified culture. The USA is an excellent counter-example, because of the many heritages and nationalities (Irish, German, Dutch, Polish, Korean, many Native American nations, many African nations, and countless other groups).
A nation can also be spread over several states. For example, the Kurdish nation is concentrated in northern Iraq, but also in eastern Turkey, northern Syria, and even Iran and Azerbaijan. The Kurds are also called a "stateless nation", because theirs is a nationality with no homeland to call their own. For another example, the German nation is predominant in Germany (but plenty of other peoples live there, too), but is also found in Austria, Liechtenstein, Switzerland, and Luxembourg.
Sometimes, a state may contain two nations in similar proportions. Belgium, for example, is populated by the Walloons and the Flemings (who live predominantly in the south and the north of the country, respectively). Sometimes there is considerable strife between such groups, as they argue about the languages on government documents and the like. Governing such countries can sometimes be a difficult balancing act.
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