Most ceilings are made of drywall. The hole left in the ceiling after you remove a light fixture can be patched using simple drywall patching techniques. The great thing about drywall is that it can be cut using a utility knife, and you don't need a lot of special tools to perform a drywall repair. After finishing the patch, the ceiling will be uniform and smooth again, and you won't be able to tell that there was ever a hole left by a light fixture. This is a fairly simple repair that can be done in a short amount of time.
Mark the area of the ceiling you want to fix. Using a straight edge and a pencil, trace a square a little larger than the hole in the ceiling left by the light fixture. It is easier to patch a square hole than a round or misshapen hole.
Cut out the drywall on the ceiling to match the square you traced using a utility knife. Tear out any leftover paper and smooth and rough edges using sandpaper.
Cut a new piece of replacement drywall the same size as your finished opening. Dry fit the new piece of drywall into the hole left by the light fixture. Your new piece of replacement drywall should fit smoothly in the hole. If it does not fit, trim any edges with the utility knife.
Attach the replacement piece of drywall to the exposed studs using drywall screws. Make sure the new piece is secure and flush with the surrounding ceiling.
Tape off the four sides using drywall tape. Cover the drywall tape with a thin layer of drywall mud or joint compound so the edges blend in smooth with the surrounding ceiling. Cover any screw heads with a thin layer of mud. It may take a couple layers. Let the mud dry and sand with fine sandpaper between coats until you achieve a seamless patch.
When attaching drywall using screws, make sure you drive the screw deep enough to leave a depression in the drywall. If you leave screw heads exposed you won't be able to get a smooth finish. Repaint the entire ceiling to achieve a finished look. If your ceiling has a textured finish, you can replicate it using special paint and rollers.
Most ceilings are made with standard drywall that is 5/8 inch thick. Make sure you use a patch piece of drywall the same thickness as your ceiling to achieve a smooth finish.