How do I sound-proof walls?

You would have to build another wall and insulate that. Anything you put on top of the old wall will have to be covered. For example if you use a 2" foam board it will have to be covered. Any added insulation will help.

Because you're renting, your landlord will have final say in what you want to do, so discuss with him, first. Unless you can locate a clause in your lease re: noise, you may have to endure this until you find another, quiter apartment. If the written "no noise" clause exists, you can bring that fact to the landlord's attention, and contact an attorney about placing rent in escrow until situation is corrected, but I'd be surprised to actually see that occur.

If it's not rented or your landlord agrees, it's best to build a 'floating' stud partition wall. A normal wall would have the studs anchored to the party wall but a floating stud would have the top and bottom plates only anchored to your floor and ceiling. With the sides in place, fixed to the top and bottom timbers, ensuring a gap between the side timber & the wall, be sure to fix a barrier to the back of the studwork to prevent your insulation falling through - a few boards of very thin ply/mdf/whatever is cheap should suffice.

Any sockets on that back wall, be sure to extend all the wiring through to the new surface.

Fill the gap between studs with a nice, thick, heavy duty insulation batt. The gap either side could be packed with loft insulation or even pipe lagging foam (it's cheap enough). then simply fix your new drywall finish, making sure to add in all your sockets (maybe even add in a few additional ones!). Be sure to finish off the tiny gap between drywall and side walls with a flexible sealant. All this should really help to dull the noise

Hang a movers blanket or quilt for an easy solution. You can paint the movers blankets to make it more appealing.

Thanks, everyone, for your help with sound-proofing my house. I own the house, so I can make the changes suggested. I think the floating wall would do the job best. This week an insulation contractor is coming to give a quote on using blown in insulation on the outside walls and perhaps the inside walls as well. I want to keep my tenants happy.