Citizenship is the status of being a citizen. If you have citizenship in a country, you have the right to live there, work, vote, and pay taxes!
Citizenship comes from the Latin word for city, because in the earlier days of human governments, people identified themselves as belonging to cities more than countries. Citizenship is more than merely living somewhere. If you have citizenship, you have a whole set of rights that non-citizens might not have. Usually you have citizenship in the country you're born in, but if you're an immigrant from somewhere else, you have to apply for it.