Many Greek city-states had slightly different setups, but their basic structure was the same: a single astu (city-center) with administrative control over a chora (territory). Athens controlled a large chora (Attica) as the result of synoekismos, or synoecism as the Brits would say. This is the process by which, usually early on in the Iron Age, various smaller population centers came together (literally "housed" together) to form a larger political unit: a polis, comprised of astu and chora. Why one center became the astu vs another was usually dependent on population, military might, religious importance, or the like. In Attica, the smaller population centers were organized into districts ("demes"). Some examples are the Piraeus (the harbor), or Sunion (a temple center in the south of Attica). These were not "cities" in that they had no independent authority in the business of Athens the polis. You should think of them as neighborhoods within a modern city, with some localized civic or religious structures.