This is a no-nonsense way of repairing sheetrock the old school way as shown to me by an old timer. The best part about it is you use no sheetrock tape just the sheetrock itself and a little bit of sheetrock mud.
Step 1: Tools and Materials
6" sheetrock broadknife (putty knife)
All purpose joint compound (sheetrock mud)
A piece of sheetrock that is roughly 1" larger (on all sides) than the hole in your sheetrock that needs repair. (So if the drywall that needs repaired is 3"×3" then you would need a piece of drywall roughly 5"x5")
120 grit drywall sanding sheets
drywall sanding pad
Step 2: Let's Get Busy!
First and foremost you need some damaged sheetrock. Ok, simple enough. Once you've selected an area requiring some attention, gather up the necessary tools and materials.
Start off squaring up the damaged drywall with your framing Square and utility knife. Once you are done with that move continue to the next step.
Step 3: Preparing the Patch
Once you're done squaring up the hole, measure it and add a minimum of 1" all the way around the hole. Mark on the backside of said patch the actual size of the hole opening. Using the utility knife and framing Square proceed to score the backside of the drywall along the edges of the dashed line. Break along the dotted lines and peel the gypsum from the paper all the while being cautious not to rip the paper.
Step 4: Applying the Patch
So once you've remove the gypsum from your patch it's time to patch the hole. First apply a liberal amount of All Purpose Joint Compound (sheetrock mud) to both the opening around the hole as well as the actual patch. Next carefully align the patch to the hole and insert it. Take a 6" sheetrock broadknife (putty knife) and smooth out the joint compound. Don't worry if it isn't entirely covered with mud you are going to need a second and possibly 3rd coat for optimal results. Let patch dry completely before moving on. Once the patch has dried lightly sand with 120 grit drywall sanding sheets. As noted above you may need a second and/or a third coat of mud for optimal results. If more coats are required be sure to lightly sand the area before applying 2nd or 3rd coats.
And that's it, pretty simple.