The laws concerning municipal corporations (incorporated villages, towns, and cities) are state laws, so there will be 50 rules. In addition, some cities are established by a city charter, which may set out special rules and name the offices. In some states, villages or towns* are organized under a different provision of state law than cities. So in Illinois, for example, the top official in a village is technically called the village president rather than mayor. Though some municipalities have "strong-council" forms of government in which the mayor has virtually no power or duties except ceremonial ones, I've never heard of a municipality in the US that doesn't have a singular executive at the top of the organizational chart.

* (To complicate the language even more, in some Eastern states, the civil divisions of counties are called towns, but those are different in organization and function than municipalities.)