They can all mean the same thing. However in terms of differences: A nation can refer to a group of people with a distinct culture, even if they are not in fact a distinct country. In the USA, some of the native american groups are referred to as nations. Some people might refer to a certain province in your country as a nation as well! A county is just that. Generally it's a self-governing entity that is independent of formal control by any other country. 'State' can be either a sub-group within a country, or it can be used to refer to some countries as well. In the case of referring to a country, it typically is used to refer to the government of that country, rather than the physical extents of the country.

Well, if you are referring to the USA, nation and country are basically the same thing and the states (like I live in North Carolina) are the 50 subdivisions of the whole country. In the states we have many cities. Each state has a capital city.