Funk and Wagnall's Encyclopedia tells us that atonality is "the absence of the system of harmonic relationships called tonality." Ummm, okay... If that's confusing to you, don't worry; we'll try and explain this.
The best way to explain the concept of atonality is to first explain the concept of tonality. All tonal music has a key, such as "C major" or "F minor", etc. All the notes in a key are related to a central note, called the tonic. As you've probably guessed, the tonic note of "C major" is C, the tonic note of "F minor" is F, and so on... The tonic is the most important pitch in a piece, and is its center of gravity. Hearing the tonic gives a sense of resolution. All tonal music returns to the tonic at the end to bring closure. For example, think of your country's national anthem (or, if you prefer, any other national anthem). Now sing it, but DO NOT SING THE LAST NOTE. You should feel an urge to just belt out that last note, because it is the tonic. All the notes in the song have led up to set up a sense of tension that can only be resolved by singing that final tonic note.
To put it simply, atonal music is music without a tonic. Now this doesn't mean that atonal composers just write songs and take out the last notes. Whereas tonal music features one pitch (the tonic) that is more important than all the others, atonal music treats all notes as though they were of equal strength. Most people aren't used to hearing music this way, however, which is why many people find modern atonal music hard to listen to.
Historically, Western music had been steadily progressing
towards complete atonalism ever since the Romantic Period. The increasing use of chromaticism during the 19th century led to
a weakened sense of key. Later, the Impressionist movement, led by Debussy and Ravel, displayed several atonal
qualities, like the use of chords for non-tonal functions. Bartók moved even further
away from the tonal tradition by generally ignoring it, except for
some instances where he used it merely as a dramatic device. An
important figure in 20th century music is Arnold Schoenberg, who developed
the twelve-tone system for composing
atonal music. This system marks the beginnings of serialsm, which is the repetition and
variation of a given sequence in any musical element.