How do I repair a ceiling where a portion is falling in and needs to be replaced?

Mark off a squared or rectangular-shaped line around the damaged area. Determine the orientation and location of the ceiling joists. Extend your squared off markings so that two opposite sides of the outline fall on center, directly over the ceiling joists. Cut out the box shape, and install new drywall into the area. Fasten along the edges falling over the joists. Also, make sure all of the damaged drywall is removed and that the problem that caused it is also repaired. Finally, finish as you would a regular drywall installation.