It is quite simple to repair a hole in a Sheetrocked wall, however there are minor differences in methods of repair depending on how large hole is.
Small size holes (1/4” or smaller)
With small holes, simply apply a little joint compound or wall putty with a small putty knife or trowel. Let dry, then sand lightly and paint.
Medium size holes (Greater than ¼” in diameter and smaller than 2” in diameter).
First clean out hole and surrounding area of any debris and loose sheetrock. Then apply a mesh tape, specifically manufactured for sheetrock applications, across hole. Use your putty knife to ensure that it is applied evenly on wall surface. Next apply a liberal amount of Joint Compound over hole and mesh. Let this set for 1 to 2 days.
After initial coat has been applied and has had time to fully dry, apply a second skim coat of Joint Compound over area. However, this time spread Joint Compound over an area that is 2-3 times size of original hole. Basically you want to flare out area that you are repairing so that it will blend in nicely with rest of wall. Again let area dry for a day.
Finally lightly sand area and apply one last skim coat over area. Again spreading Joint Compound out a little further than last application. Let Joint Compound set up one more day, sand lightly and then paint.