Having the latest and greatest tech can sometimes come at a price. For instance, subwoofers can cause quite the ruckus if not controlled properly. Investing in a couple of foam panels can help avoid a situation where neighbors are banging on your walls every time you fire up a movie.
You probably don't need industrial grade materials to get the job done, but it's not a bad idea to check out your local home theater shop to see what's available. The normal price range for a single sound panel is roughly $80-$160 each.
An alternative is to build your own using materials found at Lowes, Home Depot, or your local hardware store. While it's not perfect, investing in a little sound dampening and sacrificing some wall space can go a long way in keeping your neighbors happy while freeing you to enjoy that lovely home theater equipment you've invested your hard earned dollars in.
(Images: Troublesome on Minhembio)

Shaw's Original Fir...
would this work on a ceiling? loud upstairs neighbor
They do make ceiling versions but Im afraid this would not help you rid your noisy neighbor :) But your music & movies would sound awesome at home!
it would absorb your noise, not the neighbors.
I want a way to reduce the echo in our open floor plan. No walls, all hardwood (except for an area rug in the living area), and 6 kids = really loud. Would these help? Jennifer
This always kills me: the word is damping, not dampening.
Unless you're installing sprinklers.
I need this to absorb MY kids' noise!
Hi Jennifer... This would help remove the echo from your kids but probably not volume. It is a lot like a carpeted room versus a hardwood floor.
Nice setup with the Rotel. Always loved their products! They are one of the companies that got me into Hi-Fi!
Another option is ceiling tiles like those used in T-Bar ceilings in offices. A friend of mine glued them to the walls of his band's practice room. They work great and look a lot better than you would think!
Are there any solutions that work in the reverse? I.e. dampen the noise coming from next door.
Looks like a pair of Dynaudio Focus 110's. Lovely. I've been selling Dynaudio for over three years now and love the line. Great for designers because they have a great Danish craftsmanship story and great for audio nuts because they sound amazing. Have no personal experience with Rotel but respect them due to their reputation. Can't see the platter close enough to see if it's a Pro-ject, Rega, or Music Hall turntable though. Nice setup! Looks beautiful, and must sound beautiful!
@Jennifer, these would help quite a bit in your situation. Just a few panels like this can absorb a lot of acoustic energy that would otherwise just bounce around the walls a few times before reaching your ears. You might notice many restaurants use small panels like this on the ceiling, and how much louder (and harder to have a conversation) it can be in the ones that don't.
Does anyone know where to purchase the brackets used to mount these sound panels to the wall? Looks like a great solution!
There's one important note to remember when dealing with sound treatment:
You can dampen reflections, but you can not significantly reduce volume for your neighbors without undertaking extremes such as building an entire insulated room within a room.
Soundwaves can be thought of as liquid, to help you understand their dispersion. You can place various surfaces to break up the direct and reflective patterns, but you can't stop "the liquid" from spreading or going through to the neighbours.
As many bands learn in their garage days, doing things like hanging carpet can actually make more noise for the neighbours. How? Because carpet attenuates high frequency, not lower, and damping those frequencies can fool the band into turning it up.
Also, don't forget that lower frequencies are far more powerful. A single wavelength at 80hz is over 4 meters long. This wavelength laughs at sound panels, and needs dedicated bass traps to even attenuate it in a manner that will affect the listening position in the room. If you want to stop sub-bass going through to your neighbors... you need to build a room inside your room. With isolation or heavy matter (such as rock wool or a lot of sand).
The upside to this, now we're thinking about "sound inside the room" rather than "annoyed neighbors", is how cheap and effective little changes can be. Bookcases are fantastic for sound dispersion (both in breaking up the reflective pattern, and the variations in density of the reflective/absorptive matter).
Building your own reflective panels are also cheap and effective. If you are keen to go down this path, have a look around your favorite restaurants and bars. Many have (and many more should have) subtle sound treatment in place. After all, a big cement space full of plates, loud talking and cutlery can be painful on its own. Add some subtle diffusion (placement of objects and reflective surfaces that break up direct reflections) and some absorption (to take in the energy of the sound waves) and you've got a livable space. You might even notice this if you live in an apartment block where some apartments have tiles/hardwood flooring, and some have carpeting. You can "hear" the difference just walking past them in the hallway.
A great little video from a friend of mine shows his own DIY treatment for his home studio space, complete with before and after audio examples.
DIY Acoustic Treatment - Before and After Comparison
TLDR? Do it to make your room sound good to you. The neighbours are another issue altogether.