Here's basically how it works: a band will pre-record various parts of their performance, usually on a computer hard drive, that they cannot perform live (for whatever reason). Such parts may include additional guitar riffs, synthesized music (like percussion, organ, strings, or horns), soundscaping, and even additional backup vocals. The band will then perform their songs live in time with the pre-recorded material. How do they coordinate the live performance with the recorded one? Usually through the drummer, who has an earpiece that plays a metronome-like "click track." The click track indicates when the backing track's music will begin, allowing the drummer to count his bandmates into the song to fit the backing track. The stage lighting is often also matched with the click track, so that the lighting effects are timed exactly to the songs.
The result is a live sound that is a full and perfect re-creation of the studio recording found on the album. This is why, for example, with only four members, U2's live performances--featuring numerous guitar parts, synthesizers, and additional vocals--sound so good. This is also why Keane can get away with having only one singer (who plays no instrument), a drummer, and one keyboard player, with no bassist or guitarist in sight. And it's not just modern, synth-heavy bands. Many traditional rock groups, such as The Who, rely on backing tracks. The undeniable fact is, for a very large number of touring acts, much of the "live" show was recorded long before the band took to the stage.
This 2001 article from Onstage Magazine, which refers to backing tracks as "the dirty little secret in the world of live performance," offers an interesting how-to on using backing tracks effectively. But isn't it cheating to use pre-recorded music when the show is billed as "live"? Doesn't the use of backing tracks mean that sometimes the band is no better than a karaoke act? Or are backing tracks simply necessary to satisfy concertgoers' expectations that their favorite bands' stage performances sound "just like the album"? The answer, no duh, is Yes.
In this performance of "Where the Streets Have No Name" (from the very important! black-and-white concert film Rattle and Hum), count the personnel on stage and then count the instruments heard. And notice the headphones on drummer Larry Mullen, Jr. who counts the band into the song. Also, try to read the subtitles. Whoever wrote those has the worst penmanship I've ever seen!