A city-state is a country that is really one city, plus the land around it, and that city rules itself. It is not a city inside a bigger country, and it is not just a capital. In the past this was a common way to live: Athens and Sparta were city-states; later, places like Venice were too. The city was the government. Today the idea is rarer, though actually people still use the word for a few small places that work this way, such as Singapore or Vatican City — one city that is also a country. If a huge modern city is important but still belongs to a larger nation, it is not a city-state.