Some time around WWII, shortwave radio users started to stumble across random radio stations whose broadcasts were just a string of numbers, being read aloud by a voice (often female). These mysterious broadcasts have been found around the world, each distinctive in its own way, each one as cryptic as the last.
No organization, either government or private, has ever admitted to running a numbers station. They are featureless, nationless, and essentially impervious to being decrypted. These stations broadcast 24/7, but they are frequently only heard at night, due to the better conditions for shortwave radio transmissions carrying farther.
These mysterious broadcasts are clearly a form of communication, but with whom? The standard speculation is that most of them are being run by governments communicating with embedded spies.
Let's say you're a British spy living undercover in Russia. Every night you tune your shortwave radio to a spot on the dial where you can pick up your assigned numbers station. You can tell you have the right station when you hear its broadcast begin with a few bars of a particular agreed-upon song, or with a specific English phrase.
What follows this station identification is a string of numbers, which you write down. You then use today's sheet from your one-time pad to decrypt it, and find your instructions.
Of course, it might raise suspicion if numbers stations only broadcast when they had new instructions for their spies. So they broadcast 24/7, and the message changes at regular intervals, and the spies need to know which day and time to listen in order to get the right message. (Or maybe all the other messages just translate to "ignore this message.") You can see how complicated the spy game can get! I would never have managed as a spy. I simply don't have the patience.
Other numbers stations are thought to be run by drug cartels. They presumably are broadcasting pickup and drop-off locations and times, or perhaps announcing the next meeting.
Numbers stations have long had an active fan base, people who track the stations, attempt to locate and identify them, and trade information about them with other fans. Some of the most long-running stations have achieved a sort of fame, like the Lincolnshire Poacher (so named because each broadcast begins with a few bars of the folk song of the same name) and the Atencion station (which begins each broadcast with "Atencion!").