We had an electrician come to do some work, and he had to move an outlet from one area to another, and do some other stuff on another portion of the wall. I'm left with a square outlet-shaped cut out in my dry wall, as well as another cut out that he filled and nailed together. How can I fix these?
The first repair is a piece of cake. Get some drywall tape and a bucket of joint compound and tape away. Rather than explain it here, just Google "taping drywall" and you'll get more information than I could ever provide.
The second one is little trickier because you'll have to support a square plug of drywall. You can get drywall clips from your home store just for this purpose. You can also get a strip on wood a few inches longer the hole and screw it into the inside portion of the hole from the outside. Then cut your plug and place it in the hole and secure it to your support with a few drywall screws, the tape like you did to the first one.
If you think you'll ever have to access the second hole for other electrical work, you can just mount a blank outlet cover over the hole.