OK, quick primer on shortwave radio. If you have any questions, please ask me.
OK so the beauty of "shortwave" or HF ("high frequency" as it's known to hams) (1-30MHz or so) is that these signals are reflected by the earth's atmosphere (a layer called the ionosphere) rather than absorbed or passed through like other frequencies. Because of this phenomenon, you can communicate with stations very far away and well beyond line-of-sight. It's commonly referred to as "skip" because the signals skip off of the ionosphere. The distance you can get depends greatly on the frequency since different frequencies will skip at different altitudes.
Most shortwave broadcasters are state-run media from different countries and are almost always using AM as the mode of modulation (modulation is the method by which information is put on a radio signal). Someone mentioned above quite correctly that amateur radio operators (hams) use SSB which stands for single-sideband. SSB is a special type of AM and requires a SSB-capable radio to demodulate. There are two types of SSB, upper and lower or USB and LSB. Typically (for no real good reason) frequencies below 10MHz use LSB and above 10MHz use USB. SSB is a narrower bandwidth signal and as such the quality of the signal is reduced. It's main advantages are that it uses up less frequency space and that it is more intelligible with a weaker signal. Another mode used by hams is CW (continuous wave). CW is the act of turning the transmitter on and off, quickly, to send morse code.
My recommendation would be to find a radio that has more or less continuous coverage from 500 KHz to 30 MHz and also SSB capable. Your traditional AM broadcast band is 500KHz to 1700KHz. Above that starts the ham and shortwave bands. Frequencies around 2-4 MHz will generally be local, maybe a couple hundred miles or so. Up in the 6-7MHz range is where some international broadcasters start. These freqs are generally good in the evening. Up around 10-15MHz, these frequencies are useful all day. Above that in the 20MHz area there is less activity and more so during the day.
Check these sites for SWL (short wave listener) listings:
Prime Time Shortwave - Your guide for English shortwave broadcasts
NASWA WWW Shortwave Listening Guide
It's been a while since I was in the market for a receiver, but there's lots out there. Try ebay and eham.net/reviews/ . Read the reviews! You should be able to find something for 100-200 bucks that is decent to get started. One thing that I did not see mentioned that is just as important as the radio if not more so is the antenna! For starting out, you could just string up a "longwire" inside a room. I'd make it at least 15ft long, but try to go more like 50 ft, especially if you can get it outside. If you do put it outside you will need to be cognizant of lightning. Best advice for a beginner is to just not leave the radio (or anything) connected to the wire when it's not in use.
So why do all of this? Well, some people like the challenge of pulling in weak or distant stations. A more practical use is to get information from another source (Russia, China, N. Korea, Iran can provide some interesting perspectives and entertainment!) Of course during an emergency you could expand your listening radius greatly from that of a typical AM/FM radio.
Like any hobby, there are different levels you can take it to. There is a lot more to the world of radio but I wanted to keep this fairly on-point so as not to discourage you from trying it out.
Please let me know if you have any questions. There are dedicated radio forums as you might imagine, but there are quite a few NES hams and radio guys too.