what makes a person a resident of a county

"Resident" can mean different things for different purposes.  Thus, for tax purposes a person could be considered a resident of one place while he is considered a resident of a different place for jury duty.  In general, a person is a resident of the place if he or she intends to remain there.  If this person is living in the nursing home temporarily, with the intent to leave it in the near future, he or she might not be considered a resident, but if he or she expects to remain there for the foreseeable future, he or she would probably be considered a resident.