Often shortened simply to "bed," in radio terminology
bed music is the music that is used behind a speaker while they are on-air. Bed music sets the tone for what the speaker is about to say, and provides subtle cues for how the audience is supposed to interperate what is being said.
For example, in the United States, many newscasts start off with something that follows a "dit-dit da-dit-dit" rhythm. Likewise, a kicker at the end of a newscast, that is, a "light" story, usually about underwear or beer, is usually underscored with lighter, "wacky" music than the headlining story.
When I worked on the radio in college, I ran a specialty show centered around international music (cf. world music), and made it a point to never stop the music. I played beds every time I went up, and I got pretty good at figuring out what works as a bed, and what doesn't.
By far and away, the easiest bed music to find is from one of these two categories. A funky little groove won't detract from your words, and it will keep the audience bopping along. The downside here is that most of these tunes tend to wander into the 4:30 to 8:00+ category, so they work best for concert listings, or short interviews, since otherwise the DJ should never be up for more than two minutes or so. Of course, you can always fade the song out and transfer into your next song, but it's easy to accidentally break your flow this way.
Music in a foreign language
If your audience can't understand the words being used, they'll pay attention to you, as they should. I had a great piece, Kansha Sabira by The Rinkin Band, played on the koto and sung in the shrill, nasal Japanese of a middle-aged woman that I liked to use near the end of the show when I was saying my goodbyes. These songs are usually closer to the ideal length of a pop song, so they work when you only have a little bit to say.
What may sound like an obvious choice is best used with caution. Although originally designed to go behind something, these tracks have a tendancy toward being garishly orchestral and over-emotional. They also can be frustratingly short, sometimes under 1:30. Just the same, there are some great little ditties by Mark Mothersbaugh out there.
Why bed music is so important
In two words: "dead air."
Dead air is the most horrible sound in the universe. If you only listen to commerical radio, chances are that you almost never hear it. The rule of thumb is that every second of dead air after the first is worth about one minute of bad music. Bad music = no listeners = nobody listening to commercials = no funding = no radio station. Bed music allows you to break this rule. It allows you to insert a dramatic pause. It is, by the way, the reason why a radio DJ will play a drumroll clip during a contest: the noise allows them to pause and create tension.
The other main use for bed music happens mainly on public radio stations. There is a rule in editing a "longer story" that something has to happen at least every 45 seconds: a twist in the story, a change in tone of voice, a new topic, something. If a professional radio reporter is doing the talking, they can usually handle this themselves. In recorded interviews with ordinary people, such may not always be the case. Enter bed music. Again, setting a mood for the monologue, bed music can cover up a pause edited into the speech. The PRI show This American Life is a consistant source of examples of this use of bed music.