Unfortunately, in most cases, nations are not groups of people under a government (if this were true they would be more properly termed nation-states). This has proven to be one of the main challenges of international relations over the last century.

A state is a legal, sovereign entity with a government and a political system irrespective of any national definitions (nations).

A nation is a perceived entity - defined by a people living with a shared heritage, language, ethnic relationship, common history, etc. Ernest Renan may have best described nation when he laid it out as a continuous emotional commitment of a distinct group which celebrates is "nationhood" with anthems, poetry, and other symbols.

As an example, I offer that (what was the former) Yugoslavia was a state of many nations: Croatia, Serbia, Kosovo,...

Another example and perhaps a clarification:

Long before the creation of the State of Israel in 1948, the Nation of Israel consisted of Jews separated geographically around the world, though united in identity.

Taltos' explanation is right on the money, but the example of Yugoslavia is a little confusing because many of the nations in greater Yugoslavia went on to get their own states. Also, I would not consider Kosovo a nation but rather a political unit dominated by members of a greater Albanian nation.

As Taltos alludes to, nations of people are not necessarily circumscribed neatly in state boundaries: a circumstance that is responsible for a lot of war and conflict. In this respect, the former Yugoslavia, Kosovo, and now Macedonia are excellent examples.

Na"tion (?), n. [F. nation, L. natio nation, race, orig., a being born, fr. natus, p.p. of nasci, to be born, for gnatus, gnaci, from the same root as E. kin. 44. See Kin kindred, and cf. Cognate, Natal, Native.]

1. Ethnol.

A part, or division, of the people of the earth, distinguished from the rest by common descent, language, or institutions; a race; a stock.

All nations, and kindreds, and people, and tongues. Rev. vii. 9.

2.

The body of inhabitants of a country, united under an independent government of their own.

A nation is the unity of a people. Coleridge.

Praise the power that hath made and preserved us a nation. F. S. Key.

3.

Family; lineage.

Chaucer.

4. (a)

One of the divisions of university students in a classification according to nativity, formerly common in Europe.

One of the four divisions (named from the parts of Scotland) in which students were classified according to their nativity.

5.

A great number; a great deal; -- by way of emphasis; as, a nation of herbs.

Sterne.

Five nations. See under Five. -- Law of nations. See International law, under International, and Law.

Syn. -- people; race. See People.

© Webster 1913.