By the time we start to see references to Homeric poems, the Homeric poems are traditional, somewhat like the bible (but without the religious fervor we might attach to the bible as tradition). Archaic Greece used the Homeric poems to illustrate points, derive rules, and the poetry was generally part of an everyday Greeks' intellectual baggage. The Greeks thought of Homer as their history. They realized that there was a discontinuity between their society and Homeric society and called the time depicted in Homeric poems and other heroic sagas "the heroic age."
In the Archaic age, the Greek polis emerges. The word polis is the root of our word "political," and is usually translated as "city-state," presumably because generally a polis consisted of a concentration of people in a little city and surrounding territory. But not all city-states were like that. Some, such as Sparta, were more a collection of villages. Many people speak of the city-state as something uniquely Greek, but it was not really so: Phoenicians seem to have had similar organizations in their colonies. Nonetheless, the "mythology" of the city-state, that is, the way the Greeks came to think of their organization, was uniquely Greek and it is what influenced later ages. What actually happened is probably a combination of factors including colonization, rising wealth, new ideas about territory, monumental building (temples), geographical situations (each valley with its plain formed a nice political unit), funeral practice changes (burial became more widely available in Athens at least). Perhaps the very act of colonization spurred on the formation of a more tightly defined governmental system that was the polis. It is clear that the Greeks exported their governmental systems to their colonies.
From about 750 on, the great intense wave of colonization occurred (there had been colonization by Greeks and others earlier too). The colonization was probably not so much caused by overpopulation as by acquisitiveness, curiosity, and ambition. In that same time, writing was re-introduced, coinage began, and stone temples began to be built (but note that Lefkandi provides an earlier instance: perhaps there were more that we have not found? Note also that they were not palaces). Formal organs and rules of authority evolved in various places at various times‹laws and regulations began to usurp tradition. There were few kingdoms, except at Sparta. In Athens, the Basileus ("King") became largely a religious officer. The best guess is that they disappeared without leaving much trace in legend, etc, largely because they were merely petty kings in the first place and their disappearance was not dramatic. The assemblies and councils, all informal in the Homeric poems, became formal. Fixed rules arose for selection and rotation of officers.
The typical Greek community was small and independent. It had an agora (marketplace and assembly ground), and perhaps an acropolis (citadel to which the people could retreat in case of war). It also had a territory that may have been clearly marked by outpost sanctuaries (some were, some were not). Most had some sea frontage, and the sea was their road-system.
In addition to the polis as an organizing unit, there were leagues (called "amphictyonies") of groups of city-states here and there in Greece. Probably there was one around the oracle at Delphi, there may have been one around Lefkandi, the Ionians of Asia Minor had a 12-city one. These larger organizations were religious and served to unify quite loosely groups of Greeks. We should not picture Greek religion as one single religion, but it nonetheless tied the Greeks together. Various versions of each god were worshipped at different places. For example, Apollo was worshipped at Delos and Delphi in irreconcilable ways. The Greeks appear to have tolerated these discrepancies well. Many poleis had some claim to religious importance, and they vied with each other. Eleusis and Sicily claimed to be the place where Demeter gave humans the gifts of agriculture. Delos and Delphi both claimed Apollo. Dodona and Crete both claimed Zeus. Athens particularly claimed Athena as a patroness. The myths gave the cities a history of sorts. The nobles claimed descent from gods or heroes, their festivals, cults, and beliefs were locally unique but tied into the overall system of Greek mythology with Zeus at the head and a group of "Olympians." As Herodotus the historian says "(Hesiod and Homer) first fixed for the Greeks the genealogy of their gods, gave the gods their titles, divided among them their honors and functions, and defined their images." (Herodotus II.53) That is Herodotus' opinion, and is not entirely correct, but it attests the Greeks' belief in their own religious unity in spite of a plurality of local traditions that were not all reconcilable with each other.
The temples that began to be built in the Archaic era were not houses of worship. They housed the cult statue and held the treasury and cult objects. They also served as centers for games (the Olympic, Pythian, Isthmian, and Nemean). In addition to temples, other arts began to unify Greece. Homeric poets traveled, and Homeric poems seem to have been pan-Hellenic. Towards the end of the Archaic era, in the second half of the 6th century, Athens captured the pottery market for most of the territory settled by Greeks: that shows a large and wide-spanning trade network. Mythological and other scenes were often depicted on the pottery from Athens, which will have further reinforced the unity of Greeks.
Note that in all of this, although there certainly were preeminent leaders and aristocratic families, the temples, the art, and the culture was not like that of a palace culture of say Babylon. There was not a strict hereditary succession, and the government and culture did not revolve around any single individual. This is a far cry from the society of Homer. In spite of that, the Greeks thought of Homer as their history. They realized that there was a discontinuity, and called the time depicted in Homeric poems and other heroic sagas "the heroic age."