Clear furniture and objects out of the area of the room you will be repairing. Since drywall makes a mess it is a good idea to cover large furniture with plastic sheeting to contain the mess.
Position a ladder under the area of ceiling that needs to be repaired. Use a stud finder to locate ceiling joists. Mark these locations in pencil and use a straight edge to show where the stud is to a few inches beyond the repair area. Ceiling joists generally run in one direction and they are evenly spaced. Find joists outside of your damaged area.
Draw perpendicular lines above and below your damaged area so that you have a rectangle or square for a patch size. Use a utility and drywall knife to remove the damaged area. Try to adjust your stud or joist line so that your cut is 1/2 the width of the joist. This will give you a joist to screw into with your repair.
Measure your opening and cut a piece of drywall to fit the patch. Since you are patching from one joist to another, you should not need to add blocking between your joists; you should be able to screw your patch into the joist on either side to hold it in place. Use drywall screws and dimple your screws into the drywall.
Cut and apply fiberglass mesh tape over the seams. Overlap your corners for strength. Apply drywall compound to the mesh and patch using a 4-inch drywall knife. Spread the compound evenly over the mesh and nail holes, and thin out the edges. Allow the compound to dry completely, which usually takes about an hour.
Sand the compound and patch it smooth with the surrounding ceiling using a hand sander with a coarse sanding mesh. Wipe clean with a rag.
Apply several skim coats of drywall compound using an 8-inch drywall knife. Allow each coat to dry completely. Sand after the final coat.