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If the hole is close to an electrical outlet or phone line, make sure you check inside it so you don't hit any wires while you work. Feel around the hole with your hands, or peer inside using a flashlight. If you find a wire, take note of where it is located and plan to work carefully around it when you repair the hole. Use a ruler and a level to measure and draw a rectangle around the perimeter of the hole, then cut it out using a utility knife or drywall saw. This will enable you to neatly patch the hole with a piece of drywall the precise size you need, rather than making an irregular patch. Cut the backer boards about 4 inches (10.2cm) longer than the height of the hole. Line the first backer board vertically along the left edge of the hole. Use one hand to hold it in place tightly while using a drill to screw two drywall screws through the intact drywall just below the hole, and two through the drywall just above the hole. Use the same technique to install another backer board along the right edge of the hole. Pine or other soft wood backer boards work well for repairing drywall, since they're easy to screw into. Be sure to hold the boards in such a way that the screws won't scratch or puncture your hands when they pop through the backer boards. Measure the thickness of the drywall and purchase a piece of drywall large enough to patch the hole. Cut it to size using the drywall saw so that it will fit neatly into the hole. Place the drywall patch into the hole and screw it into the backing boards on either side, spacing the screws 6 inches (15.2cm) apart. Most hardware and home goods stores sell scraps of drywall in different shapes and sizes. Look for one large enough to patch your hole so you don't have to buy a complete sheet of drywall, which will probably be more than you need. Load a joint knife with joint compound and apply it to joints, the seams where the patch and the wall connect. Quickly apply paper tape to the joints and use a taping knife to smooth the tape into place, making sure there are no bubbles or lumps. Apply a second coating of joint compound and let it dry. You can add a little water to the compound to thin it out, making it easier to spread and feather out along the wall for even blending. Be sure to remove excess compound as you go, so that the transition between the patch and the wall is as smooth as possible. Pull the taping knife in one direction. Laying the tape evenly can be tricky. It's worth starting over if you lay it crookedly, since the tape is important for blending the patch with the wall. Once the first few coats are dry, smooth the edges by sanding them with a fine grit sandpaper. Fill in any gouges and uneven areas by applying another thin coat of compound. Let that dry, and continue sanding and adding more compound until the surface is even and smooth. Wait at least 24 hours between sandings. The compound should be completely dry, or you may create more ruts and gouges instead of smoothing the surface. After the last sanding, use a primer to get the area ready for painting. When the primer is dry, paint the area using the same brush or paint applicator you originally used to paint the wall.

Summary:
Check for wires. Cut a rectangle. Add backer boards. Install a drywall patch. Tape the joints. Sand the area and add another coat. Prime and paint the area.