Move any furniture and other obstacles away from the area where you will be working. Cover the floor and anything close to the hole using the drop cloth. Cut the drywall around the hole using a utility knife to make the hole larger, exposing the studs or joists. The reason for this is that you will need to have wall studs or ceiling joists available to attach the new piece of drywall.
Measure the opening of the hole using a tape measure. Cut a piece of drywall to the size of your measurements by scoring it using the utility knife. Snap the piece off the main sheet of the drywall. Place the piece of drywall into the hole and secure it using drywall screws, the screw gun and screw bits. Drive the screws in until they dimple the drywall slightly. Do not over-tighten or they will go all the way through.
Place a thin layer of drywall compound around the seams of the patch you installed. Place the drywall tape over the seams and cover it with a thin layer of drywall compound. Tear the drywall tape at the end of the seams using the putty knife. Allow the drywall to dry for 24 hours. Put on the surgical mask and safety glasses before you begin sanding the work area.
Sand the area smooth using the sandpaper and sanding block. When drywall compound dries, it will shrink, so you might have to add another coat after sanding the area. Paint or finish the wall as desired after the patched area is dry and sanded. When repairing a hole in the ceiling, use the stepladder to access the work area. Gather up the drop cloth by picking it up by the corners to avoid allowing the drywall dust to fall off it. Clean up the area as necessary and put the furniture back into place.