This depends on which county you're talking about. Every state is different, and within some states, every county, city, or other municipal entity can differ, to boot.
Generally speaking, municipal entities (which includes counties) are legally extensions of the state government - so at some level the state legislature is in charge of redistricting insofar as it decides who to delegate it to, if they do delegate it.
Those states that do delegate it, for example, Washington, have to provide for the method of such divisions in the authorizing statute. Generally speaking, they will require the municipality to define the process in its chartering document. (Basically the Constitution, but for the municipal entity.)
Beyond that, you'll have to be far more specific.