The county administrative offices are usually located in the county courthouse located in the county seat of the county. The elected representatives who run the county are named differently in different regions presumably as dictated by state law or the county charter. In some areas, it is the county executive. In New Jersey they are known as freeholders. In Missouri as judges of the county court.
The function of the county varies from state to state. Most at least operate a recorder of deeds and probate office. Most have a sheriff, a jail, a county highway department, and often parks, social services, unemployment services, etc.
In the original thirteen colonies, the counties are usually divided into municipalities which may be either boroughs or townships, depending on population. Most of other states have cities which are incorporated and have their own administrative functions, but areas outside those boundaries are unincorporated and are administered by the county.
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