A nation (1) is a group of people joined together by language, culture and ethnicity. While today most (major) nations possess an independent state, this was not always so; the rise of nationalism is the 18th and 19th century was when this idea (that each nation deserved its own state) became influential. There are still today however many nations without states, such as the Kurds.
In another sense (2), the word nation also refers to a territory under a single independent government and the inhabitants of such a territory; in other words, a de jure or de facto state.
A certain amount of flexibility is needed when interpreting this definition. Is the European Union a nation? De jure it is not, though it might be considered to be one de facto.
Is a city a nation, when it has its own local government? No, because it is not independent. (Though again, it might be considered to be one 'de facto' if the central government was very weak.)
Can't we just use a United Nations list of countries? No, since we'd like to refer to nations that disappeared before the UN was created, or de facto nations that are not recognised as such by the UN. (Whether or not there can be de jure states not recognized by the UN depends on whether one accepts a constitutive or declarative theory of statehood; which one is correct is controversial at present in international law.)
"Nation" is also used in other cases which don't meet the definition above:
- native American usage: compatible with sense (1) above.
The nations that combined to form the United Kingdom are sometimes still refered to as such, even though self-government is currently limited. This usage is justified as an instance of sense (1) above, and also historically since once centuries ago each was independent.
- In international politics and map making, national boundaries are not redrawn immediately when a government loses control of part of its territory. Multiple groups may hold power within a single "nation" for long periods of time.