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,...