My girlfriend knocked a chair over when she tripped and it made a large hole approximately 6 inches by 6 inches in a triangle shape. How do I fix this as fast and as cheap as possible ?
There are hole patch kits that can be purchased at most hardware stores. If that's not an option (maybe the hole is too big?) you can take a small square of drywall that's bigger than the hole and hold it up over the hole, mark around the piece with a pencil and then cut the hole out to the outside of the line. Then take two slats of 1x3 and use screws to attach them through the wall on either side of the hole creating a backing for the patch. Once that's done apply the patch to the backers and proceed to mud and tape the seams. Here's a diagram:
If you need help with the mudding/taping process there are other q/a's on this site that cover it better than I can so I won't go into it.
Get some thin wood strapping. Cut two pieces a bit longer than the hole. Slide them through the hole, and drive wallboard screws through the surviving plaster to hold them in place. Cut a wallboard patch to fit into the opening, and drive wallboard screws through it into the wood supports you've just installed. Spackle the joint between the patch and the plaster; spackle over the screw heads. Smooth the surfaces you've spackled. If necessary, lightly sand the area and/or apply more spackle and/or alternate the two until the screws, and the joint between the old and new wallboard, are completely level. Prime and paint.
Less traditional solution: There are self-adhesive screening patches available. Get one larger than the hole, stick it in place, apply spackle over and through it to create a new plaster surface. Smooth (giving particular attention to hiding the edges of the patching screen) until it looks sufficiently level (ie, not an obvious patch, even with raking light at a low angle). Prime and paint.
I'm sure there are other answers; these are the two I've used. Both are cheap and fairly easy. The traditional solution is probably more durable, but I can't prove that.
Remove all of the damaged sheet rock with a razor knife, or small saw. Once you have a hole with solid edges, measure the thickness of the sheetrock on your wall. Buy, beg, or borrow a small piece of sheetrock in that same thickness. You can buy small pieces at your local home center, but if any new homes are being built near you, you can get a piece just for asking. Cut the replacement to the same size as your hole. The more precisely your patch matches the hole, the better. Make the edges smooth with no small pieces of facing paper on the edges. Sand the facing paper off one inch in on both your patch and the existing wall. Get a big balloon. Stick it in the hole and blow it up. This will give you something solid to work against. Place the patch in the hole, use sheetrock tape and mud to patch the seam. There are lots of youtube videos, and the folks in the paint dept of your local home center can also be valuable help. Two or three coats of mud, sand after each, then paint to match. Take a piece of the broken sheetrock you removed to your home center, and they can match the paint on your existing wall.