The political structures of these new peoples were sporadic. They lived in poleis, or city-states, and had hundreds of varying structures.
Out of the midst of these poleis emerged Athens, between 800 and 400 B.C.E., with the first example of democracy in human history. In addition, the highly militaristic society of Sparta evolved alongside it, and these two states rose to be Greece's biggest powerhouses.
Greece produced many important ideologies that would last throughout the ages. They were polytheistic, however their focus was more on worldly affairs. This focus, called secularism, worked much like Confucianism did for China, and led to many of Greece's great thinkers.
Among the like were Socrates (focus on ethics and human nature), Plato (described Socrates forced suicide for conflicting human nature with religion/tradition), and Airstotle (all fields, tutored Alexander, Mathematical symbols).
These achievements and focus on Natural Law led to the era from 500-300 B.C.E. being classified as the "Hellenic culture."
With these successes under their belt, Greece was not ill-prepared when the Persian Empire arrived and demanded their allegiance. The many poleis then united, creating a West vs. East mentality that is still applicable today. This united Europe for the first time, and Greece held off Persia, and sparked their decline.
However, Athens attacked Sparta's allies shortly after, and initiated the Peloponnesian Wars (431-404 B.C.E.) between the two states. During the next 100 years, Macedonia, to the North, created a large military force and handed it off to Alexander the Great, who would conquer most of the known world in just 13 years.
He conquered from Greece through Persia, and when he died, he would divide his land up between his 3 favored generals. In this division, the Hellenistic culture spread throughout most of the known world, consequently allowing Greece to vanish in the years with it.