Cut the damaged area out of the ceiling if it isn't already removed. Drill a hole in the surface so your saw can fit inside, and then cut out the damage, using a jigsaw.
Place the cut-out piece on a new panel of drywall and use it as a template to draw the outline for a new section.
Cut out the new drywall. Use a utility knife to slice along the drawn lines.
Measure the width of the opening in the ceiling. Cut 1-inch wood strips to 3 or 4 inches longer than this length, and slide them inside the hole. Place one strip at each side of the hole. If the opening is wider than 18 inches, cut a third strip for the middle. These wood strips are what you will use to attach the new drywall, but first you must attach the strips to the ceiling.
Drill drywall screws through the drywall along the edge of the opening and into the strips.
Place the new drywall in the hole and up against the wood strips. Drill drywall screws through the new piece and into the strips.
Cover the seams around the repaired hole with drywall compound, and then place drywall tape over the compound. Use a taping knife to spread on the compound.
Smooth on a second layer of compound over the tape, and let it dry for at least 12 hours.
Sand the surface of the compound to smooth it out. Cover the repaired surface with another layer of material, and thin it down as you spread it away from the seams. Let the surface dry once again, and then sand it smooth.
Paint the ceiling repair area to match the rest of the surface. Use a paintbrush and matching paint.