Identify the cause of the damage and fix any leaks or moisture issues first
Remove loose drywall, peeling paint, and damaged tape
Cut out the damaged section into a clean square or rectangle
Measure the opening and cut a matching drywall patch
Install backing strips or support cleats behind the opening if needed
Screw the patch into the backing strips or ceiling joists
Apply drywall tape over all seams
Cover the seams with joint compound
Let the compound dry completely
Sand the dried compound smooth
Apply additional thin coats of joint compound as needed
Sand between coats until the surface is level
Prime the repaired area
Paint the ceiling to match the surrounding surface