Sound Proofing 101

In an apartment this is really difficult. If you can get the book "A compendium of materials for Noise Control" from the library check it out. It's out of print, but can be found at many universities. What you will find is that noise travels through the structure almost as much as the walls, particularly bass frequencies. Thus you need to isolate the area with either a large mass, or a broad airgap. The best is to use a combination of both. Sound isolation chambers essentially float a room within a room. Obviously, this is extreme and would not be appropriate in your circumstance, but what I'm getting at, is that there really is not a very good solution short of building another wall with an airgap between that wall and the existing wall. This is assuming that the frequencies are on the low end and penetrating through the structure. One interesting thing that could be done (to find out if I'm right about that--not knowing the construction I could very well be wrong). If your neighbors are patient enough, so an SPL plot accross the frequency spectrum inside your room and pretty loud 85 to 90 db--or close to the listening level that is bother some (assuming this won't damage any of your equipment), then plot this in the neighbors, you can see what the attenuation coefficients are for each frequency. If it's higher frequencies--which I doubt. Owens corning 705 fiber board mounted on the wall will help a great deal (it will deaden the room). If the fequencies are below 300 Hz--what I suspect. You would just about have to build a wall inside a wall.

Rives gives excellent advise.
Also if your speakers (or stands) are spiked into the floor, then try spiking them (via disks or coins) onto concrete paving slabs, laid on the carpet. This should provide a steady base while reducing coupling with the building structure.
IF this all fails I'd really consider getting a good headphone system. Senn600s plus an inexpensive amp ala Creek would allow you more freedom to enjoy the music.

If you are thinking of putting a covering on the wall, a company named Celotex makes some sound reducing material that is meant to be installed behind the drywall, but in your case, you could probably install it over the top of the drywall. We built a box out of the stuff and put a little boom box in there and it did a very good job at reducing the sound. You might want to look into it. It is fairly inexpensive.

I once was about to rent a house with one wall of my future sound room to be common wall with the neighbor. I did a fair amount of research on this problem.......

The best way to keep the sound inside your room is with a VERY HEAVY fake wall. There are several ways to do this. Far and away the cheapest way is to build a wall inside. (2X4's with sheetrock on each side.) Then fill the wall space you have created with sand and pray the sheetrock doesn't break open.
I didn't say it was easy, just cheap.

Richard
PS: I didn't rent the house.

This really requires two things. First, sound travels through everything in the room...not just the wall. It travels through the floor, the ceiling, any common airducts, etc.

Your best solution will probably be to make a wall, with a gap, as mentioned. The best way to do this (seeing you are using a full range audio system, would be to make a concrete cinder block wall using Helmholz resonators. You can get these from Proudfoot, they call them the Sound Blox. While not extremely cheap, they are very reasonable. If you cannot use these, use cinderblocks, and after the wall is completed, cut vertical slices through the outer wall of the cinderblocks every 24 inches or so. These act as a makeshift helmholz resonator. Then, apply drywall to the outside of this. Make sure to leave a gap of at least 6 inches between the existing wall and the new one.

If you want to go all out, then you need to float the floor as well. After your wall is complete, get some neoprene floor covering (expensive), and lay it atop of your existing floor, up to the new wall. Then put a layer of fiberboard subflooring and plywood flooring. Top with carpet, and you're set. In a residential area, there's not much you can do about the ceiling, but with the wall you should get an STC rating of about 40 (40 db down on the other side) and with the floor STC 50-55. Hope this helps!

Brian