Measure the hole, and cut a piece of drywall that is at least 3 inches wider and longer than the hole.
Mark the hole's dimensions onto the backside center of the patch, then draw an outline that matches the hole. Score along the outline with a utility knife to cut through the back paper. Extend the scored lines straight out to the outside edges of the patch.
Snap the drywall along the lines, breaking the gypsum core but leaving the white face paper intact. Carefully peel the core of each waste piece away from the face paper. When you're done, you'll have the hole-shaped patch in the middle surrounded by 1 1/2-wide flanges of face paper (viewing from the backside).
Spread a 1/8-inch-thick layer of mud in a 2-inch (min.) swath around the ceiling hole, using a drywall knife. Apply a liberal coating of mud to the edges of the patch center. Paste the patch into place and support it with one hand while smoothing the face paper with the knife. Work the knife outward from the center, pressing it firmly to squeeze out excess mud from behind the paper. Let the mud dry.
Apply two more coats of mud to feather out the patch so it transitions smoothly into the surrounding surface. Sand the final coat smooth, if necessary, or apply a spray texture (see Tips) before painting.