what is the purpose of a nation?

You need to differentiate between a nation, a country, and a state.

A nation is a group of people who identify themselves as a common group. It serves as a way to identify oneself as well as provide a community on which an individual can rely on for support.

A country is a physical area under the jurisdiction of a government.

A state is the organization that runs the country and represents its interests. This is why we call it the United States - back then each colony considered itself a separate state that agreed to join the Union. (It was also by this reasoning that the Southern states seceded and formed the Confederacy.)

Most countries today coincide with a nation (the "nation-state": one state for one nation), but there are exceptions - more in the past (The Austro-Hungarian empire was one state that held many nations - Austria, Hungary, the Czechs, Bosnians, Romanians, etc.), but some today as well (Koreans consider themselves one nation under two states, North and South).