So, demes as administrative units were generally a feature of Athens, although some other cities also had them. At Athens the demes replaced the tribes as the minor administrative units after Cleisthenes' revision of the constitution: at Athens a tribe (φυλή) was formed from taking one of the ten "thirds" (τριττύες) of each zone, City, Coastline, or Interior. The τριττύες themselves were composed of a varying number of demes, of which there were in Attica 139. The tribes were generally the more important administrative units, but the demes had a certain share of organization. Though the tribes formed the basis of distribution of seats within the βουλή (500 members chosen by taking fifty from each of the ten tribes) the distribution of members within each tribe was supposed to be done proportionally with the size of the demes it contained. Each deme elected a local δήμαρχος (which, amusingly, is the word that later authors use to describe the Roman tribunate), whose duties mainly involved registering new demesmen, as well as some other minor administrative functions. Other than that the demes operated kind of like miniature versions of the city. Demes held individual assemblies, and could appoint whatever officials they thought were necessary as they saw fit. Demes were in charge of organizing their own festivals and maintaining their own finances as they saw fit, with whatever means had been voted on in their own assemblies. Some demes even set up inscriptions documenting their public activities, e.g. Eleusis

Other cities had radically different methods of lower-level administrative organization. A fair number of cities used demes (I think it's like 28 known cities, something like that) but many did not, and of those that did we generally don't have the documentation to compare them to the Athenian system following Cleisthenes. Many cities maintained their older tribes--at Corinth, for example, the eight tribes formed the basis of all minor administrative duties