I've got a wall of these in my listening room, make the room sound much better. Not sure how great they are for keeping sound out of your place, that wasn't an issue for me. Might be worth a shot though.

http://www.mioculture.com/store/pc/viewPrd.asp?idcategory=2&idproduct=9

You could hang some heavy fabric type panels on the offending wall.

book shelves filled with books may help as well.

Cork flooring panels. Use them on the floor or on a wall or two.

http://www.acousticalsurfaces.com/acoust_ceilings/ss_ceiling.htm
This stuff is pretty good, comes in 1" and 2" thicknesses and you can use it as a tack board. Compared to some of the "real" acoustical solutions, it's also pretty cheap.

Rather or not it's aesthetically pleasing, well that's up to you. I've seen the stuff in person and think the charcoal color is actually pretty nice, has a nice texture, and could be a cool backdrop to some nice art.

having as many soft surfaces as you can will help, even if it's not directly blocking the noise from coming in. the noise will get absorbed instead of bouncing around the room.

I dig the cork wall idea.

I should print this out and slip it under the door of the woman who lives above me and walks in High Heels at 2 AM on hardwood floors.... which happen to be right over my bed.

it's not clear if you are trying to keep sound out or in. some of the suggestions above may help with keeping your sound in, but not with your neighbors' sound out. it seems like acoustical engineers agree that sound travels through joists and empty cavities and not just through the wall, so before investing any money to caulk or add sheet rock to an existing wall, you should do some research or get an estimate from a contractor. good luck.

Thirding the cork idea- it's a huge sound absorber. You want a material that will absorb sound waves instead of reflect or transmit them. Some DIY home recording studios use pegboard from home improvement stores for a similar effect. Fabric drapings also help.

Watched an episode of Holmes on Homes. He installed some drywall (QuietRock QR-525) that was equal to 8 layers of standard drywall. Here's a clip that I found online:

http://www.youtube.com/watch?v=ZlAZsYp2M-U

People make great sound-absorbers. We're basically just bags of water, and suck up all the energy from the vibrations.

So always have friends over for drinks.

Or go to wally's.

So far, not a single response directly addresses the original concern.

The solution should be
1) able to keep neighbor sounds out
2) temporary, since it is a rental
3) essentially DIY
(and considering 2 & 3...)
4) cheap

I would love to hear easy, temporary solutions as obviously installing new drywall or windows in my rental apartment is not an option...

QuietGlue is 20% cheaper than GreenGlue. It will be a better option.

I'm sure you think it's a better option. After all, you work for them...

Your most cost effective solution will be to find another place to rent.

Soundproofing Company sites offers some good insight to this type of noise problem. To get remarkable results you will need to go to some extreme lengths, but if your apartment is in the right place at the right cost, then it might be worth putting a months worth of rent into it. I read this post
Soundproofing solutions for walls. There are a bunch of stuff I didn't even think about on their site. I learned to think of my room as a fish tank. Look for the leaks. There are a ton. The question is how far do you want to go.

I decided to installed Green Glue in my rental and it was worth doing. The landlord still doesn't know but I did all the drywall work myself. I wasn't that hard. Well worth the money for feeling like I finally have some privacy!! Side note: I'm on year two of my lease.