The power of the county government varies widely from state to state as does the relationship between counties and incorporated cities. In New England, counties function only as judicial court districts (in Connecticut and Rhode Island, they have lost even those functions) and most local power is in the form of towns.
Many states have counties named after U.S. Presidents such as Washington, Madison, Polk, Jefferson, etc. Counties are also commonly named after famous individuals, local Native American tribes once in the area, cities located within the county, and land features (Cerro Gordo, Iowa, meaning "Fat Hill" in Spanish).
Lists of counties by state: