How did ancient civilizations organize their society?

Answer 1

Sparta was a Kingdom. The Spartan Kings however were mostly leaders of the army; their other tasks were mostly ceremonial. Sparta was effectively ruled by a Council of Elders. The other (mostly) city-States were oligarchies, ruled by members of the most powerful families. That is even true for Athens. Although Athens is usually hailed as "the birthplace of democracy", that democracy existed only for a limited period in its history. It moreover took the form of a limited number of mass meetings (usually carefully orchestrated by the ruling families) on matters of great national importance. You could say they had the occasional referendum; there were no democratic institutions and all the executive jobs remained in the hands of the oligarchy that ruled the city.

Other aspects of Greek society: slavery was everywhere and it formed the backbone of the economy of the Greek societies. Freeborn male citizens were the only ones with legal rights. With the exception of Sparta (where women lived on practically equal footing with men) Greek women were seen as the mere property of their fathers or husbands and were allowed very little freedom of movement.