For small holes up to approximately ½ "you can simply use spackling compound and a putty knife. Apply the spackle, wait until it dries then sand smooth. Spackling compound will shrink a bit so sometimes you will need to apply a second coat after the first coat dries, you will then need to sand it again before touching up the paint.
For medium holes 1/2 "to 3" you will need to get a drywall patch (there are various other methods to repair a hole this size but a drywall patch kit is the simplest). Most of these kits come with a patch that is self adhesive. Simply cut the patch so that it is larger than the size of the hole by about 1 "and stick it over the hole (some kits actually enable you to stick the patch through the hole then pull it tight behind the back of the hole enabling you to fill in just the hole).
Then you will need drywall compound (which can be found in smaller containers than a gallon) or spackling compound. Using a wide blade putty knife (3 "or wider), apply the compound over the patch and a couple of inches surrounding the hole, then after it dries, sand it smooth. You will need to repeat this step a couple of times until you can no longer see where the patch is, sanding after each coat dries. Finally, touch up the paint.
For larger holes 3 "and larger it is typically best to replace a section of the drywall. Find where the closest stud is, then cut out a rectangle around the hole including half of the width of the stud. Then, using the cut out piece as a template cut a fresh piece of drywall to fit the hole. Place the drywall in the hole and use drywall screws to attach it to the stud. (it is even better if you cut out a piece that is large enough to expose half of two studs so that the drywall is even firmer in place. Studs are usually positioned 18 "apart but could be anywhere between 12" to 24 "apart.)
Using you "mudding knife" (or large putty knife), apply a thin layer of drywall mud to cover the entire joint. Next, apply paper drywall tape over the mud and using your blade, embed the paper firmly into the mud, scraping off any excess mud. After this dries apply another 2 to 3 coats of drywall mud over the joint tape extending approximately 6 "on either side of the joint, sanding between each coat. Once the repair job is smooth you can touch up the paint.