Measure the length of the square hole the vent came out of, using a tape measure. Cut a 1-by-3 inch board into two pieces, using a saw. Each piece should be 6 inches longer than the length of the hole.
Place one strip on each side of the hole. Pull the strips firmly down, against the drywall, and screw one drywall screw in each end with a drill.
Cut a new piece of drywall large enough to fit in the hole. Place the drywall into the hole and secure it to the strips with drywall screws. Run a strip of sticky mesh drywall tape over all the seams.
Apply a thin layer of drywall mud over the seams, using a 3-inch taping knife. Hold the knife at a 30-degree angle and spread the layer as evenly as possible. Wait for the mud to completely dry.
Spread a second layer of drywall mud over the first layer, using the 3-inch taping knife. Wait for this layer to dry. Spread a third layer with a 6-inch taping knife. Feather the mud on the edges to blend into the old drywall.
Sand the mud smooth and blend it into the old drywall, using a sanding sponge. Paint the new drywall and mud with primer and a paintbrush or roller. If your ceiling has texture, use a texture spray to give the new drywall texture. Paint the patch the color of the rest of your ceiling.