Fixing a hole in the wall
How large is the hole exactly? If it's 16" or wider I would consider removing some drywall so that you could reattach the new drywall straight to the studs.

If the hole isn't that large, normally you would use two pieces of furring strip that will be 4 inches longer than the hole in the wall. The furring strip will serve as makeshift studs.  Evenly space them apart so that the drywall will be stable when installed. 

Measure 2" down from each end of the furring strip and mark it. When you place the strip inside the hole, align the strip so the mark you made is even on both sides. Using drywall screws, screw through both sides of the wall 1" from the mark you made (remember, you have 2" to place with). Repeat with the second furring strip. Once the strips are securely attached to the wall, screw the new drywall to the furring strip (use at least four screws).

Finish by applying compound or plaster (whatever your finish is), sand if necessary and paint.