This Instructable shows how to repair a hole in sheetrock. This method works for holes from about 1/2 inch to 2 inches across.
Smaller than 1/2 inch, just fill it in with spackling or sheet rock mud.
Larger than 2 inches, use the method of a board behind the patch.
I'll show you the method that I use. I like to use joint compound rather than spackling, but that is a choice for you. The pictures don't show the actual sheetrock mud, just the method used.
Step 1: Make the patch
Use a piece of drywall 3-4 inches bigger than the hole you are repairing.
Turn the patch over so that you see the back of it.
Use a drywall knife to cut a "tic-tac-toe" pattern on the back of the piece.
The center square should be larger than the hole you are repairing.
Bend the pieces away from you and remove each piece, leaving the center square intact.