If I am suing someone do I sue them in my county or in their county of residence?

If you are both NC residents, you may sue in either.

The answer is yes.  In most cases, there is not just one answer to this question.  For example, if you are a New York resident who gets into an automobile accident with a New Jersey resident in Connecticut, you can probably bring suit in any of the three jurisdictions.  Without knowing the details of your case (for example, is it based on a contract which contains a clause limiting where you can sue for its breach, is it based on a tort which occurred in a particular county, etc.) I cannot be certain, but in most cases jurisdiction and venue would be proper in the county of residence of either the plaintiff or the defendant.