I can answer this question in terms of what happened to Mycenaean society, as it's an area that I've briefly studied before, and I still have some of my books on the topic.

By the 13th century BC, Mycenaean society revolved around "palace economies" in which the wealth of the land governed by each palace was stored and then redistributed out from this central location. These palaces were built similar to the Minoan style, but were fortified and usually built in defensible locations, unlike those found on Crete.

These Mycenaeans who inhabited the Peloponnese, and the areas around what were later Athens and Thebes, were not politically united. Each palace was its own political entity containing records inscribed on clay tablets. This is important because many of the surviving tablets are around because of fires which hardened the clay during the collapse of the Bronze age in Greece.

Beginning around 1200BC, many of these palaces and their administered towns and other population centres were attacked, destroyed, or abandoned. Many of these palaces were never reoccupied thereafter, some recovered however, only to collapse again soon after. Some palaces continued on in the form of towns which existed near the ruins of their respective palaces, including Mycenae and Tiryns. But by 1100BC, the Mycenaean system no longer existed.

To get to what life was like in that period, a contrast with what it was like before was necessary. Directly after the collapse of the Bronze age, population in Greece dropped to the lowest that it had been in thousands of years. Now that towns and other population centres could no longer rely on the redistribution of wealth largely in the form of agricultural goods which the palaces were responsible for, surviving population centres had to become politically and economically self sufficient. The "qasireu" of the Mycenaean age which had been responsible for administering towns slowly became what were known as "basileus" which translates to "king". These figures were the leaders of Greek communities during the dark ages.

An example of life in one of these new communities is the village of Nichoria. Abandoned in 1200bc, it was repopulated by 1075 as a village cluster with a peak population of around 200. At the centre of the village was the Basileus' house. Which was a long hall which was rounded at the end. These houses likely acted as the centre of life of communities in the dark ages. The construction of this particular example was packed earth and mud brick, with a thatched roof.

Although there was a general decline in economic activity during the dark ages, it was not a time of total stagnation. Farmers kept farming, herds were still tended to, and weavers kept spinning wool and flax. It was during this time that the use of iron first started to appear.

Citations

Pomeroy, Sarah B. A Brief History of Ancient Greece: Politics, Society, and Culture. 2nd ed. New York: Oxford UP, 2009. Print.

http://www.arch.ox.ac.uk/classical-archaeology-periods.html

EDIT: Formatting.