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Patching drywall is a task that nearly every homeowner will eventually have to do, whether they're moving into an older home, hanging pictures on the wall, or preparing to sell a house and needing to patch the holes left by pictures.
For a simple patch, you can use spackling compound, which comes in many container sizes. For a simple hole, you don't have to do anything except push the compound into the hole, smooth it off, let it dry, and apply a couple more coats later as necessary. As it dries, spackling compound shrinks a little, so a filled hole will look a little less filled after it dries. Spackling compound doesn't shrink quite as much as joint compound can when you're doing a spackle job like this.
For a medium-sized hole, such as one near the baseboard or one caused by a doorknob hitting the wall, you could use heavy-duty joint compound tape, like mesh tape or plaster tape, placed right over the hole. For this, it's recommended to use multiple layers of tape, perhaps three heavy coats. The mesh tape is the same kind installers use to tape the seams when they first put drywall up on the wall. Heavier-duty tape is a little thicker and is made for patching.
There is also a pre-made patch with an adhesive backing. It is metal and comes in three or four sizes. The mesh overlaps the joint, so you can lay it on the wall, where it will stick, and then apply layers of compound or spackle over it.
Joint compound is what installers use to cover seams when they first put the board up. For this kind of patch, spackling compound is the one to use. The difference between spackling compound and joint compound is that spackling compound is generally a little thicker. You still need to try to make the first coat as smooth as you can, go over it again with another layer while keeping it smooth, and hopefully sand it after the second coat. If not, apply a third coat, make it smooth, and then sand it.
For a bigger hole, where something was made through the wall, one approach is to cut a patch to go over the damaged section of the wall. You cut out the damage, but make the patch just a little smaller so it fits in. You could trace the patch onto the wall and then make the hole bigger to match, rather than trying to get a precise patch. The problem is that there is nothing to fasten it to, so you would need to put something behind the wall. You could slide a board behind the opening, put a couple of screws in place, and then remove the center screw and screw the patch to the wall. There are also clips that can be installed on two or four sides, depending on the size needed. You put the clips in, then put your patch in with a couple of small screws, going through the flange behind it. Flatten those spots with a hammer so the patch lies flat enough that it won't show.
Another way is to take a piece of drywall, cut it bigger than the hole, and remove the perimeter. You cut the patch to the size of the opening, but the paper remains as the tape. When you put it on the hole, make it a little smaller so it fits easily, since it has some play in it. First, apply glue or compound to the wall around the perimeter. In this case, use spackling compound for a hole this size. For anything bigger, use joint compound. Put down a quick layer, press the patch in so it holds in place, and then use your knife, held upright, to scrape away the excess compound from the seam. Let that first coat dry. This coat holds the patch in place, and it won't go anywhere. You don't need any tape, and you won't get cracking. Wait for it to set before applying the top coats. Keep the first coat as smooth as you can, apply a second coat and keep it smooth, then a third coat, and sand as needed. If you do this well and sand and feather the patch properly, your eye will never see it, and you won't get any cracks.