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Short waves are radio waves transmitted in certain bands and frequencies. The most interesting feature they have is propagation: an atmosphere layer above the Earth bounces back these waves, allowing the broadcasts to be received at high distances. The Wikipedia entry on Shortwave radio details it probably the best. Don't forget to read the entry on shortwave listening (SWL) also.
A handy guide from Bill McFadden gives us the full details on the short wave chart. Briefly, it's between 2300 and 26100 kHz where it's most interesting to listen with a common shortwave receiver. Also, Eton has an interesting shortwave tutorial available, a good introduction into the bands and listening methods.
2300-2495 kHz 120 Meters; 3200-3400 kHz 90 Meters; 3900-4000 kHz 75 Meters; 4750-5060 kHz 60 Meters; 5850-6200 kHz 49 Meters; 7100-7350 kHz 41 Meters; 9400-9900 kHz 31 Meters; 11600-12050 kHz 25 Meters; 13570-13800 kHz 22 Meters; 15100-15800 kHz 19 Meters; 17480-17900 kHz 16 Meters; 18900-19020 kHz 15 Meters; 21450-21850 kHz 13 Meters; 25600-26100 kHz 11 Meters.
Of course, you need a shortwave receiver - portable or not. Which one to get is a topic highly debated - try reading the specific pages on swling.com, dxzone.com, dxing.com. Try not to get the cheapest one, as you'll probably be disappointed in quality and forget the hobby quickly; look for SSB, PLL synchronisation, a digital display and most important of course: reception quality. I went for a Tecsun PL-660 coupled with a chinese Kestrel W31MS active loop antenna which gave pretty good results: personal record so far is receiving a station at 12800 km away (see movie on the left).
You can get some good deals on Amazon and eBay, so make sure you're following those sites (Amazon also has customer reviews which are a good source of information regarding the reliability of radios). Another excellent source for equipment reviews is eHam.net.
You might wonder what's so interesting in listening to radio? after all, it's full of FM stations for music, we have Internet access, etc. Well, it's something special in receving a very distant broadcast full of crackles and pops (as it's analogue after all and not digital). It's true, if you're too young you might not know what I'm talking about :-) One major point to consider is that shortwave radio cannot be filtered like the Internet (ok, it can be jammed, that's true, but it's much more difficult). You need just batteries and a cheap radio to tune in on a frequency; this means it's the primary source for millions of people living in areas of the world where there is no electricity or Internet access.
Number stations are mysterious shortwave stations which broadcast weird things: series of numbers, strange sounds or songs, encrypted messages. Most famous one is probably UVB-76, nicknamed "the Buzzer". It's assumed these stations are used by various countries' governments to send messages for their spies in foreign territory; to receive these messages the spy would need just a radio which wouldn't attract so much attention. Check out the Wikipedia entry on numbers stations, Priyom.org, SpyNumbers.com and of course The Conet Project.