According to legend, Athena, goddess of wisdom and war, and Poseidon, god of the oceans, were in dispute over which one of them was to be the patron of Athens. It was decided that they would offer gifts to citizens, and Athens would choose its desired patron. Poseidon offered mastery of the sea, its military power and its trade. Athena, however, offered the olive tree, a symbol of peace and prosperity. The city chose the olive tree, and thus it was named Athens.
Athens was one of the strongest city states of Ancient Greece. But apart from being a seat of military power, Athens is usually credited as having been the first ever democracy in human history. It was ruled by a large regularly elected government of citizens. It also held referendums on all important issues. There were referendums every week.
However, Athenian democracy excluded many demographics. In fact, only male Athenians who had undergone military training could vote and (when they were slightly older) participate in government. This left out women, slaves, foreigners, children and certain people whose rights were suspended (for legal reasons e.g. debt).
Regarded as the second most prominent Ancient Greek state and often considered Athens main rival, was Sparta. A city named by a ruling son of Zeus, Lacedaemon, Sparta is famous for the importance it placed on its military. And despite its small population, it did achieve remarkable victories.
But Sparta was actually a mixed system. It vested supreme power in an assembly of elected citizens called The Apella. Thus it was the first democracy before Athens. Also, although women were not allowed to fight, they did run the country while the men were away fighting. They had much more power and many more priveleges than in Ancient Athens. Despite this, women didn't form part of the assembly.
The Apella was only a third of Spartan rule. Sparta retained its two Kings of different lines (although they were largely symbolic) and had a few oligarchs. Together they formed the Gerousia (council of elders). The assembly could return and reform proposals that the council of elders put forward. They were not expected to greatly exercise their power and any proposals they rejected could be re-submitted after time.
Which was more democratic?
Both states used slaves and discriminated against women. But where The democratic part or the Spartan government had limited power and was mostly presided over by the non-elected part, Athens had dispensed with kings and oligarchs altogether. Every aspect of governing Athens was carried out democratically and every decision was voted upon.