So there is no coordinate data (x,y)? What are you trying to map? What is the data (census data about income, locations of bathrooms, county boundaries, counts of serial killers by county, etc...)?
If you have county data in spatial format (ie ArcGIS polygon feature class) you can use the General -City State Country type of locator and just use the County Name as the city name. You could also use the General -Gazetter and put the County name in the Place Name. There is also the General - Single Field where County name occupies the KeyField.
Take a look at: About the geocoding tutorial—Help | ArcGIS Desktop