Atonal music is music that is written with the avoidance of a sense of key. This is usually accomplished by the repeated use of dissonance.
Classical music began to dissolve conventional tonality in the beginning of the 20th century, and this idea was imported into jazz music.

A more literal definition of atonal from a compositional standpoint would be "without relying on the tonal center that diatonic harmony provides."

In diatonic harmony, a C major scale revolves around the C. With atonality there is no starting tone or baseline to work from. And I meant baseline, not bass line. Wiseacre.