Well, the first rule, of course, is that with 50 states, there are probably 70 rules. As noted by friedo, the primary political division within a state is a county. (In Louisiana, the equivalent political entity is a parish and in a couple of states–possibly Alaska and one other–the equivalent is a borough, but parishes and boroughs serve the same purpose with different names. This will, of course, be confusing when one encounters a state in which a particular variety of city organization is called a borough, but go back and reread my first sentence.)
Counties are the primary entity responsible for maintaining police protection and roads and such. They have no power to enact laws, but they may establish ordinances for the regulation of things like traffic or the prohibition of open fires that come under the jurisdiction of police and fire protection or road and traffic control.
In many states, there is a subset below county called a town in the East and Northeast and a township in the midwest that may have further derived powers to regulate building codes and such. (Beginning with the original Northwest Territory, Ohio, Indiana, Michigan, Illinois, and Wisconsin, the territories were surveyed into townships, originally squares five miles on a side, later six miles on a side and the townships were surveyed in mile-square grids. This practice has carried over into some, not all, more western states.)
This is the general layout of the original government plan. However, people do not tend to live diffusely spread across the landscape, they often clump into hamlets, villages, towns, cities, and metropoles. Recognizing that people generally prefer local control, most states have rules permitting cities to take back some of the authority initially granted to counties. In fact, if the city is large enough, it may be permitted to enact ordinances that would usually appear to be the prerogative of the state. At the lower end of this spectrum of communities, each state must decide at what point a community can begin usurping county authority. Usually the distinction made is between incorporated or unincorporated, and is usually predicated on a set limit of population.
Thus, the county seat of Geauga County, Chardon, is large enough to be an incorporated village (and I do not know the minimum size that Ohio sets for incorporation). Thus, they are able to control their own destiny regarding village streets (whereas the townships must get approval from the county in order to change rights of way), sewers, building codes, etc. To their frustration, for each of the last few decennial censuses, Chardon always falls a couple of hundred people short of the magic number of 5,000 which would allow them to petition to be recognized as a city, granting them more powers of governance and taxation.
I have used real-world examples of the sort of things that might affect cities, counties, incorporation, etc. in order to give a flavor of what is going on. However, my post will be followed by multiple posts pointing out the differences between Ohio and various other states (and, perhaps, differences within Ohio for regions that were settled at different times and fell under different Federal or state charters when they were surveyed).