Repairing a drywall hole is a job that all homeowners and even renters face at one time or another. The good news is that this is an easy DIY job that will save the labor money that a handyman would charge. Keep in mind that this is not a five-minute repair job if done properly but the results are well worth the time invested.
The first step is to make that ugly hole a more manageable shape. This is easily done with a drywall saw; just cut a square around the hole. In many cases, a larger area will be easier to work with than smaller ones. Before cutting, mark the square off using a carpentry square and a pencil. If this cutting leaves a stud exposed the job will be easier to complete. In this case simply cut a piece of drywall to fit the space and either nail or screw it to the stud. Most hardware stores sell 2' X 2' X 5/8" pieces of drywall for smaller projects.
Otherwise, cut a length of wood (a shim from the door section of the hardware store works well) two inches longer or taller than the opening. Secure it inside the opening by screwing through the drywall on two sides using a cordless drill with a Phillips #2 bit and coarse-thread dyrwall screws. Nailing is not recommended. Now simply screw the drywall patch to the wood strip, taking care not to push too hard. Two screws should do it.
Next, it's time to tape and float the seams of the patch. First apply drywall mesh tape. Don't use the older paper type; it's harder to work with and doesn't have the adhesive backing. Use a narrow taping knife to spread drywall compound over the tape. After allowing this coat to dry, sand it lightly and apply another coat; the idea is to visually eliminate the seams. If another coat is needed then just repeat this step; it is better to spend more time on this step than look at an ugly wall every day. Keep in mind that in most new construction, the floating goes through three iterations.
Now it's time to match the existing wall texture. In most residential situations, this will be an orange peel texture which was sprayed on during construction. Small touch-up spray cans of this product are available in the hardware store. They are expensive, but the good news is that most jobs require less than one can. For other custom texture patterns such as a stomp or a knock-down, use a paint brush to dab drywall compound and the taping knife to shape the surface to match the existing wall. This may take some experimentation to get right but remember that this part is more of an art than a task.
Finally, to finish repairing a drywall hole, paint the patched area. In fact, this is a good time to paint the entire wall to make sure everything matches (and ticks one more thing off the honey-do list). High traffic areas will not match well so a complete paint job is recommended.