About a hundred years ago I worked on a drywall crew. Yeah they treated me like a girl no matter how well I did.
So fixing holes and dents the boys knocked into walls was one of my jobs.
My primary tools and materials for this were a carpenter's square, a utility knife, a church key can opener, a roll of tape, some mud and a putty knife, a scrap of sheet rock, and a handful of drywall screws. Oh, and a screwdriver to drive the screws in.
What actually got used depended on how big the hole was, dents mostly took as much work because I usually had to turn them into holes.
Step 1) square up the hole with the square, the utility knife and the church key
Step 2) measure off and cut a piece of scrap about a 3/4 inch to an inch and a half bigger than the hole. (Depends on how big the hole is)
Step 3) make a hole in the middle of the scrap big enough to hang on to.
Step 4) mark off and start screw holes around the hole (these days I would use a drill for this), within the excess margin you cut on the scrap.
Step 5) Fit the scrap into the hole, holding on by the hole in the middle, and screw it into place. Screw the screws in a bit deeply so that the heads are below the paper. Back then it took a fair bit of elbow grease, make sure you support the scrap firmly. If it is a big hole, you may want to make your hanging on hole big enough to put your whole hand through to support the screw hole from behind.
Step 6) cut another scrap of drywall to just fit into the squared off hole.
Step 7) mud the site liberally
Step 8) fit the new square of drywall into the mudded cavity and tape the edges, making sure to cover the screws, and sponge the edges.
Step 9) go do some more holes while this one dries.
Step 10) When completely dry sand off as usual.
With new materials out there these days, I imagine that for a one off you can buy a kit that is easier than this old school method.
I got along well on this job. It was a union site and I was being paid the same as everybody else, but some of the guys thought that that was just so wrong.
So I wound up getting laid off because they caused a fuss with management.