Run a stud finder over the ceiling, and locate and mark the center of the ceiling joists on each side of the damage. Run pencil lines down the center of those joists--you will now have two parallel pencil lines. Connect the lines at each end, at least 6 inches on each side of the damage; this will result in a square or oblong shape.
Run a tradesman's knife several times along all the lines, until the drywall is completely cut through. Remove the damaged area.
Measure the hole in the ceiling, and cut a new piece of drywall to that size--cut this piece out with a tradesman's knife or drywall saw. Insert the new piece into the hole and screw it to the ceiling joists--every 6 inches--with 1/5/8-inch drywall screws and a screw gun.
Cover the drywall joints with mesh tape: place it sticky side to the wall and push it with your hands to ensure that it is stuck firmly in place (it can be cut with a scissors, or knife).
Pour some joint compound into a drywall tray, and place some compound on one side of a 6-inch drywall knife. Smear the compound over the mesh tape, and feather it out--this means applying a layer of compound onto the ceiling past the edge of the tape, decreasing the thickness of the layer as you go. This will help the eye not see that the compound is thicker on the tape than the surrounding areas. Apply compound also to cover all screw heads. Wait for the compound to dry overnight.
Sand the dry compound to remove any ridges or bumps--if any screw heads or tape can still be seen, apply another layer of compound, wait for it to dry, and sand a second time.