Spray the ceiling around the hole with water, misting it until the texture is slightly damp.
Scrape away the dampened texture, removing it in an area spanning at least 2 inches wider than the hole.
Sand the ceiling around the hole with a drywall sander when the ceiling is dry.
Cut a piece of drywall mesh tape large enough to cover the hole and position it over the hole. The tape is self-adhesive.
Scoop up 1/4 cup of premixed drywall compound with a 6-inch taping trowel and smooth the compound over the hole, pushing it through the holes in the mesh tape.
Let the compound dry. When it's dry, it will turn from gray to white.
Sand off the surface of the patched hole with a drywall sander. Apply another coating of premixed drywall compound. This time, use only enough to cover the patched area thinly and let it dry.
Re-sand the patched area smooth.
Apply texture that matches the rest of the ceiling. This might be the most difficult part if you don't know how the original texture was made.
Experiment on a scrap piece of drywall before applying the texture to the patched area.
Thin drywall compound to the consistency of cake batter and roll it over the patched area with a paint roller.
Texture the wet compound by patting with a brush, swirling with a trowel, whisk broom or wallpaper brush, or by knocking off just the tips of the rolled-on texture.