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 the 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 the hole and surrounding area of any debris and loose sheetrock.
Then apply a mesh tape, specifically manufactured for sheetrock applications, across the hole. Use your putty knife to ensure that it is applied evenly on the wall surface.
Next, apply a liberal amount of Joint Compound over the hole and mesh. Let this sit for 1 to 2 days.
After the initial coat has been applied and has had time to fully dry, apply a second skim coat of Joint Compound over the area. However, this time spread the Joint Compound over an area that is 2-3 times the size of the original hole. Basically, you want to flare out the area that you are repairing so that it will blend in nicely with the rest of the wall. Again let the area dry for a day.
Finally, lightly sand the area and apply one last skim coat over the area. Again spreading the Joint Compound out a little further than the last application. Let the Joint Compound set up one more day, sand lightly, and then paint.
Holes larger than 2 inches
With large holes, I recommend cutting out an area such that a new small piece of sheetrock can be applied to the 2″x4″ studs. For example, I would consider putting in a new 16″x16″ piece of sheetrock for a hole that is greater than 2″ in diameter and less than 16″ in diameter.
The best method to remove the piece of damaged sheetrock is to use a Carpenters knife (razor blade) and score the area where you want to cut out. Repeatedly apply the knife until you have worked your way through the sheetrock. When complete, you should have about ½” to ¾” of the 2″x4″ stud showing on each of the two exposed studs.
Note: Care should be taken to ensure no wires are cut behind the sheetrock when cutting out the damaged piece.
Next, cut a piece of new sheetrock to the size of the hole, and secure it with sheetrock screws or ringed sheetrock nails.
Next, tape the cracks with the mesh tape material mentioned above. Then as described earlier, apply three skim coats of Joint Compound, with each successive coat getting flared out further. Make sure you wait for the Joint Compound to dry before applying the next coat. Perform final light sand after the final coat has been applied and has had time to dry, and then paint.