One important thing to understand is that power is decentralized in the American system. Power is pushed to lower levels as much as possible. The federal, national government has very limited powers. States and cities have more power.
Because it is not centralized, there is not one answer to your question. There are a variety of levels smaller than the state, and in each state they work differently. New York has counties that contain towns that contain villages. Virginia has counties and cities - cities are not contained in counties. Connecticut has no counties at all, only towns. California has counties that contain cities, but some areas are in a county but not in a city (unincorporated areas).
The leaders of these levels of government vary as well. I would point out here that another important concept is separation of powers. At the national level there is executive (President), legislative (Congress), and judicial (courts). These smaller levels of government also have separation of powers. In smaller cities there may be a council of 3-10 people who are the executive, rather than a single Mayor. And the legislature can be ~300 elected citizens, or even the entire voting population. So many citizens do not live under a mayor.
And separation of powers prevents any one person becoming a king. Taxes are not changed by the mayor, but by the local legislature. Cities usually only collect one kind of taxes, taxes on land/property. The city only provides a few services. And a tax increase often has to be approved by voters first, limiting how much they can collect. The police do work for the mayor, but their budget is set by the legislature, and their conduct is reviewed by the courts. In almost every city this works well enough to prevent major corruption or autocracy.
In some larger cities the city provides many more services, and collects more tax dollars. With increased money and power, the chances for corruption increase. That is why in several historical examples you see a "small king" developing, like in New York City and Chicago in the 20th century.
The conduct of mayors is reviewed by higher levels of government, so it is possible for truly corrupt mayors to be arrested by the state government and go to jail. New York and Illinois have both sent high level politicians to jail.