The Greeks had a lot of different kinds of governments, because there were many different city-states in ancient Greece, and they each had their own government. In addition, people's ideas about what made a good government changed over time.
Aristotle divided Greek governments into monarchies, oligarchies, tyrannies and democracies, and most historians still use these same divisions. For the most part, Greece began by having monarchies, then oligarchies, then tyrannies and then democracies, but at each period there were plenty of city-states using a different system, and there were many which never did become democracies or tyrannies at all.
In the Late Bronze Age (the Mycenean period), between about 2000 and 1200 BC, all Greek city-states seem to have been monarchies, ruled by kings. Homer's Iliad, and Greek mythology in general, shows us a whole series of kings like Agamemnon and Theseus, and some of their palaces have survived for archaeologists to dig up.
After the Dark Age, though, only a few Greek city-states still had kings. Sparta is the most famous of these, though actually Sparta had two kings, usually brothers or cousins, at the same time. One would stay home and the other go off to fight wars.
Most city-states in the Archaic period were ruled by oligarchies, which is a group of aristocrats (rich men) who tell everyone else what to do. Then in the 600's and 500's BC a lot of city-states were taken over by tyrants. Tyrants were usually one of the aristocrats who got power over the others by getting the support of the poor people. They ruled kind of like kings, but without any legal right to rule.
In 510 BC, the city-state of Athens created the first democratic government, and soon other Greek city-states imitated them. Even city-states that weren't Greek, like Carthage and Rome, experimented with giving the poor people more power at this time. But Athenian democracy did not really give power to everyone. Most of the people in Athens couldn't vote - no women, no slaves, no foreigners (even Greeks from other city-states), no children. And also, Athens at this time had an empire, ruling over many other Greek city-states, and none of those people living in the other city-states could vote either. Of course it is a lot easier to have a democratic government when you are only deciding what other people should do.
(And many Greek city-states kept oligarchic government, or tyrannies, or monarchies, through this whole time).
Then in the 300's BC, Greece was conquered by Philip of Macedon, and all of Greece began to be ruled by him as their king (in theory he was only leading a league of Greek city-states, but really he acted like a king). Athens and other Greek city-states still kept their local democracies or oligarchies for local government, but bigger decisions were made by Philip, and then by Philip's son Alexander the Great.
After Alexander died in 323 BC, Greece became a kingdom ruled by a series of Macedonian kings, until it was gradually taken over by the Romans between 200 and 146 BC. From 146 BC on, Greece was a province of the Roman Empire. Even after the Roman Empire in the West collapsed, Greece was still part of the Eastern Empire. In the 1100's and 1200's AD, parts of Greece were taken over by Normans, who built castles and ruled as kings.
And finally, in 1453 AD, the Turks took over and established Greece as a province in their Ottoman Empire; there was not very much change in the system of government from the Roman Empire.
