"What is involved in repairing this? Major electrical work?"

No, hardly any electrical work at all.

It's mechanical work, to securely attach the box to the wall (and move it to be flush with the wall). There may be screws or nails that have come loose; if they are inside the box then you will have to unscrew the outlets to get at those screws, then reattach the outlets to the box (this is probably the extent of electrical work, and even this won't require even detaching any wires). Otherwise the box may have a bracket extending up & down from the box itself, and that is where you would attach it to the wall.

The problem here is that what you need to get at is behind the plaster wall. Most of the work will be in busting out part of that wall to get access to it, and then repairing the plaster wall, sanding it smooth, and then repainting it. Drywall work, not electrical work.

You might try unscrewing the outlets and stretching them out on their wires, to see if it's possible to work from inside the box and run long screws into the nearby stud to secure the box in place. That would be the easiest.

Otherwise, get ready to do all the drywall & painting work as mentioned above.

Note that if the outlets themselves are secure, not moving around in the box when you plug & unplug cords (instead the whole thing moves as one unit), then this is not a seriously dangerous situation -- you don't have to fix this immediately. Just be careful when plugging or unplugging cords until you get it fixed.

I'm not an electrician, but I've replaced all of the switches, receptacles, and even some breakers in my house -- so I know my way around a bit.

I definitely agree with the above post, but I'd like to add a couple of points that I don't think were stressed.

1- Make sure that before you even take off the cover plate you kill power to the device by opening the circuit breaker. Confirm that the power in the receptacle is dead by using a tester. Then, cautiously remove the cover, and again use a tester to test each individual wire that comes into the box. Though typically frowned upon in electrical work (and I think it's a code violation) -- sometimes you may have multiple circuits that exist in the same box -- meaning just because you turned off the circuit breaker that feeds the receptacle, you may still have hot wires in the box! And again, test EVERY wire -- even the white neutral wires, which sometimes are used as hot wires and aren't marked correctly. Once you've confirmed that all power in the box is dead, you can safely proceed.

Granted, drywall work won't necessarily involve doing anything with the electrical wires. However - I've had instances before where a bare ground wire touches the stripped end of a hot wire because of how loose everything is in the box and how much everything moves -- this can cause injury and/or fire. Sadly, it can also typically be prevented by putting a couple of wraps of electrical tape around the device, but in my experience this is very rarely done -- even by professional electricians. So again, killing power before doing anything is very important.

2- If you decide that you want to replace the device while you're doing the work, carefully pull the device and all wires out of the box as far as possible, then take a few pictures of what you have. Sometimes it's as simple as a single hot and single neutral wire in the box, but it could also be as complicated as having several black, white, and even red wires in the box, along with several splices (wirenuts). To make absolutely sure that you rewire everything correctly, pictures help a ton! Another tip- is the receptacle controlled by a switch on the wall? If so, check the little tabs on the sides of the device that can be broken off. The tab on the hot side (brass screws/black wires) is likely to be broken. Pay attention to this, and do everything the same when you put in the new device.

3- This sort of goes along with my first comment, but again -- a loose device inside a box can be a very dangerous situation. Wires can come off the device entirely if they have come loose over time, then short out when they contact each other. This can cause fire/injury. But even if they're secured, sometimes in that big rats nest of wires inside the box, that bare copper wire can come into contact with things it shouldn't - and again, that can cause fire/injury. Take loose devices, even switches, very seriously. They won't get better on their own, and in fact they're guaranteed to get worse.

I really hope you didn't listen to either of the other two posters! All you need is a Remodel box (Sometimes called and old work box - less than $2.00 at any home improvement store), simply shut the beaker off, disconnect the wires from the outlet (take note of what goes where), pull the exisitng box out since it is loose, route the wire(s) through the new remodel box, set in place and tighten both screws in the box, this will cause the "wings" to retract and grab hold of the back of the sheetrock, hook your outlet back up and turn the breaker back on. Done, no need to rip your walls apart - people like that give forums a bad name, let's do this right.