If you live in an apartment, you've probably have some noisy neighbors from time to time, or lived facing a busy street. You have also maybe watched Star Wars on full blast only to get the cops called on you. Aside from tearing your hair out or getting a visit from the boys in blue, what can be done? Quite a few things, actually.
In layman's terms, sound is a form of energy, and can do about 3 things: be absorbed, reflected, or transmitted through a medium.
So, what materials can be used to help us keep sound out, or keep sound in?
Porous and fibrous materials are maybe the most economically accessible option: they reduce noise by having the sound energy enter the material via a multitude of openings and holes, vibrate the fibers, and convert sound energy into kinetic energy and heat.
Items that diffuse the noise, as well as reflect/stop transmission are of equal importance, but probably more work intensive, and expensive.
Here are some off the shelf items you can use:
Heavy Drapes - They do a great job of not only absorbing sound, but reflecting transmission out of your room from busy streets.
Sound Absorbing Foam - These are stellar options for placing around your home to mitigate noise from a concentrated source. They excel at getting rid of high frequency noise, so they won't eliminate Skrillex base drops, but they'll do about 75% of what you need.
Diffuser Panel- While the above two items cover two parts of acoustical treatment of a room, you can't neglect diffusion. They help with reducing the apparent intensity of noise off a reflected surface, and can give a room a spacious feeling. Some of them have really stylish designs, so they can also be looked at as art pieces.
Viscoelastic Material- Sometimes noise can propagate from things that vibrate. Get your mind out of the gutter, we're talking about air conditioners, dish washers, or if you have them in your apartment, washers and dryers. The noise emissions from these appliances can all be reduced by placing a viscoelastic material on them, reducing the noise transmission at lower frequencies. Just be extremely careful on placing the material where things get hot, as heat can get trapped, and possibly fry your appliance.
If you're feeling a bit creative, and willing to put the work in, an extremely effective measure at reducing noise transmission would be to sandwich the viscoelastic material between two metal plates. Normal metal sheets will ring when struck, indicating vibration and noise transmission, but this will not.