Look at the ceiling to determine the type of tools you might need. If it is a popcorn or knock down ceiling texture, then you need a paint sprayer. If it is a swirl pattern with lots of lines, then you need a trowel and wire brush. And if you see a thick, smeared on pattern, you need a trowel.
Tape a plastic drop cloth to the wall and set up a practice panel of drywall against it. Use painter's tape so that the drop cloth peels off easily when you are through.
Practice spraying or smearing on the drywall texture compound. For spraying you need to mix the compound with water to a consistency of thick paint.
Spray the compound over the surface, adjusting the nozzle for the best match to the ceiling. If you're aiming for a knock down technique, then the spray needs to produce thick beads of compound. Scrape over the top of the spray to knock down the beads flat. Use a trowel and just skim the top of the spray.
Practice spreading on compound then brush a wire brush through it to create swirls. If you prefer a stippled effect, tap the wire brush straight into the compound. Spread compound on with a trowel in various swirls or sweeps of the tool. Create thick to thin strokes to make the pattern. Continue practicing the technique required for your ceiling type until it is a close match.
Cover the repaired area on the ceiling with drywall compound in the manner required for the technique. Cover up the area, as you practiced, to match the surrounding surface. Step back and look at the patch area from a distance to see it better. If it doesn't match, you can scrape it off while it's still wet and try again.
Leave the patch area to dry for a day before painting it to match the rest of the ceiling.