Atonal music is a generalizing term used to define music that seems to lack a clear tonal center. Nearly all music in the western classical tradition is considered 'tonal': that is, its harmonic structure is primarily triadic and hierarchically organized around a prominent tonal center. Atonal music works tend to deny or expand this notion by using alternative structural strategies (frequently - but not exlusivelly - mathematical, the most famous being serialism). As a result, many listeners used to traditional tonality may find atonal music very challenging at first, in particular because the lack of a tonal heirarchy means that highly dissonant chords are far more common (and, indeed, early practitioners of the style often deliberately used these chords as to avoid any implied reference to a tonic). That said, others argue that atonality is simply another musical language which, like all other languages (including traditional tonality), cannot be learned or appreciated until one has been immersed in it.
A brief history of atonality 
Composer Arnold Schoenberg is generally seen as the first composer to fully embrace 'atonality', although a number of other composers (such as Bartok and Scriabin) had been moving in that direction for some time. In 1925, after experimenting with free atonality, he composed his Suite fur Klavier, the first piece of music ever written using the twelve-tone method of composing. His students, Alban Berg and Anton Webern, followed in his footsteps by composing numerous twelve-tone pieces; the method was generalised during the 1950s to include other parameters such as rhythm and volume, the resultant system being known as total serialism.
Contemporary situation 
All these influences over the last century have meant that it is now very difficult to draw a clear line between atonal and tonal music: for example, the music of Philip Glass has a clearly defined tonal center (and is generally seen as tonal), but it does not use harmony in the traditional way (as a means of giving the music direction), instead using repetition to create something more static; similarly, the music of composers such as Alfred Schnittke or John Adams combines elements of tonal and atonal music freely and undogmatically; this approach can be seen as far back as Berg's Violin Concerto of 1936, which sounds almost tonal in many places despite its use of the twelve-tone method]].