Damaged walls come in all shapes and sizes. From the smallest scrape in the paint to replacing entire sheets of drywall. What I'm going to attempt to cover is small damage from the size of a ring to a larger area like the circumference of a Tim Horton coffee cup.
Ring size area of damage:
For a first class repair here's my recommendation. You can purchase a small piece of drywall at the hardware store for less than five dollars. Your only going to use a very small amount as a filler, but it's always handy to have in your garage. Cut off a small piece, peel off the outer paper, a push as much of the drywall material inside the damaged area trying to fill it. You will see videos online where people may use pieces of newspaper or something else, but these paper products don't have the strength to create a lasting repair. Place some repair screening mesh over the area, this can be purchased in rolls or packaged kits. Now your ready to put a very light coat of drywall compound over the damaged area. Let dry over night, sand with finer grit sanding block until desired finish is reached. Your next step is to primer the repaired area, then your ready to repaint.
Larger damage areas:
Do you still have the piece of damaged drywall? Is it still in one solid piece? If you don't then your going to have to make a trip over to the hardware store. Cut a piece of drywall to match the general shape of the damaged area. Your going to want the repair piece to be slightly smaller than the damaged area. If your lucky and have a wall joist behind the damaged area you can drill a drywall screw right through the repair piece you just cut and into the wall joist. If not you should use a small piece of wood and use it as a brace. In effect 3 drywall screws should secure it. One screw through the drywall patch into the centre of the wood, next the other two screws that will brace the wood in place, drilling the drywall. Be aware of live wires and other obstructions inside the wall. Once repair piece is firmly in place, my choice is a mixture of sheetrock and drywall compound to create a solid base coat. Use very sparingly, if you over apply you will have a heck of a time sanding down. Next use drywall compound to build out and blend to the surface of the wall. Apply coats as required and finish with fine sanding. Primer & paint till completed.