Knock away all the damaged ceiling plaster with your hammer. Continue pulling down material until you reach the ceiling joists on either side of the damaged area.
Mark out a square area surrounding the damaged area, with two parallel borders of the area running along the centers of the joists at either end.
Use your level and pencil to mark out a square area around the hole, with two sides of the square sitting at the centers of the two joists on either side of the area. Cut out the square using your utility knife to score the plaster repeatedly until it comes down, leaving a square hole in the ceiling that's bordered on two sides by joists.
Measure the square and cut out a piece of drywall to that dimension, minus 1/4th inch on all sides. Cut the drywall by scoring and snapping it with your razor knife.
Hold the square piece in the space. Shoot drywall screws through the drywall everywhere there are joists behind it, putting in a screw every six inches or so.
Press mesh drywall tape over the four sides of the patch. Spread joint compound over the tape, using your drywall knife, so the compound spans the patch and the surrounding ceiling.
Let the joint compound dry about five hours. Sand it lightly, then spread on another layer of joint compound. Repeat the process to apply three layers, getting the final one as smooth and flat as possible. The ceiling is now ready for painting.