I finished filing and paid what I was owed; but surprisingly I got a refund from both state and fed. My return said I owed $$. So what happened? What error did I make?

Did you have a complex return ?
Did you file a return that showed 1040 Line 4a non-zero but less than Line 4b?

You should have received a letter explaining why you received a refund instead of having to pay.  If this was a check you received, do not cash it until you get the letter explaining.  If you do not get a letter in the next few days you should call the IRS to find out why you received a refund. 

There was no explanatory letter. One check already cashed. Was there an error in TurboTax calculations?