Me! it's ! try to be as detailed as possible.
what are the different types of municipal governments, what responsibilities do these governments often have, and who grants power to these local governments?

Local government in the United States refers to governmental jurisdictions below the level of the state. Most states have at least two tiers of local government: counties and municipalities. In some states, counties are divided into townships. There are several different types of jurisdictions at the municipal level, including the city, town, borough, and village. The types and nature of these municipal entities varies from state to state.

Many rural areas and even some suburban areas of many states have no municipal government below the county level. In other places consolidated city-county jurisdictions exist, in which city and county functions are managed by a single municipal government. In some New England states, towns are the primary unit of local government and counties have no governmental function but exist in a purely perfunctory capacity (e.g. for census data).

In addition to general-purpose local governments, there may be local or regional special-purpose local governments,[1] such as school districts and districts for fire protection, sanitary sewer service, public transportation, public libraries, or water resource management. Such special purpose districts often encompass areas in multiple municipalities. As of 2012, using the Census Bureau's definition, there were 89,055 local government units in the United States.[1]

Large municipalities have basic types of government, which amount to 3. These are mayor-council, council-manager, and commission.

that is alot i do not no