Patching a Small Hole in Drywall
Using a level, and straight edge, draw a tight rectangle around the hole. Using a razor blade, or drywall saw, cut a clean hole. A saw will allow you to remove most of the material, and the razor will cut it cleanly. Then you can use the hole as a template to cut out the patch.

Before the patch can be installed, you will need to install a backer board to prevent the patch from falling inside of the wall. You can use just about any piece of wood as long as it is thick enough to hold a drywall screw without splitting. The wood needs to be long enough to be securely attached to the existing drywall on 2 opposite sides for strength. The screws must be at least an inch away from the edge of the hole so it does not crumble. You can also put a screw in the middle of the backer board prior to installation to give you something to hold onto while it is being installed.

Then screw the patch to the backer board to hold it in place. You can then clean up the edges of the drywall pieces to fix any imperfections. Use drywall compound to fill in the cracks, and cover up the screw heads. Keep this coat very thin, and just put in enough to fill in the voids. Let this dry, and then tape the edges with drywall tape. After this, you can skim coat the whole area to blend in the patch.