Did I just create counterfeit bitcoins? I had 3 Bitcoins stored on my PC and I had a backup image of my PC. I restored my PC on another computer from the image and I had 3 Bitcoins on that computer also. I spent 5 Bitcoins as a test, and it worked. What just happened?
It's like two people holding on to the same $20 bill and each of them saying, "see, I have 20 dollars!".

Bitcoins can be spent/accessed by anyone that has the private key to unlock them. If two computers have the same private key stored, then they can both access the bitcoins. If one wallet spends the bitcoins, the other wallet will show them as spent as well, just like one person spending the $20 bill means it is gone for the other person.