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Numbers stations are radio stations that, not surprisingly, send a lot of numbers.
They are usually found in the High Frequency range (HF) approximately 2-30Mhz, often referred to as the "shortwave band". Very long-range communications are possible in the HF range given enough power and suitable atmospheric conditions.
Generally speaking, only governments have the wherewithall to construct really high-powered broadcast facilities, so it is reasonable to assume that these transmissions are governmental in nature. In fact, it is believed that they are messages to spies in other countries.
Okay, you're sitting at your computer which can communicate with anyone in the world who has internet access. And you're thinking "why would anybody use something as old fashioned as shortwave radio to send messages to spies?". Of course, it's possible to accomplish the same thing using computers, and I'm quite sure it is being done... a lot. There are plenty of sophisticated way to get messages out that aren't even noticiable and are encrypted so that they can't be read even if you discover one.
There are a couple of problems with this, though.
With shortwave radio:
A couple of times the U.S. government has caught spies receiving messages broadcast from Cuba. So it's a known fact that at least some of the stations are broadcasting spy messages.
The Israeli spy agency MOSSAD used to be one of the most active groups around, sending coded messages out just about 24 hours per day. Last year those messages disappeared, so perhaps they have found a better way.
Numbers stations use spoken numbers or phonetic equivalents for letters, morse code, and various types of data signals.
There is a group known as Enigma-2000 that tries to catagorize and keep track of these stations. Their website is a treasure trove of information and sound samples. Google 'em.