Atonal Music begain at the beginning of the 20th Century with the movement of what is known as Serialism (which Wagner kicked off with his 'Tristan Chord')
The 2nd vieneese composers, Berg, Shoenberg and Webern really started and developed the movement along with others like Stravinsky.
Anyway Serialsim is devided up onto the folloing series (or note rows) Melodic Seriesl, Tonal Series, Atonal Series, Symetrical Series,All-Interval Series, Symetrical All-Interval Seties, Short Series and Long Series. Tonal Series, Atonal Series,
Composers write a 'Note Row' (or a 'Tone Row' as it's sometimes known. Note Rows are constructed within the Total-Chromatic (that is all the twelve notes contained within the octave. They can be in ANY order the composer chooses but must write them for a purpose, not just any old series can be written. If he wants to use his row for melodic purpose, then he will construct his row to have melodic intervals. Even if a Tonal Series is composed, it will more often than not, still be atonal.
It's such a massive subject that it takes a lot of explaining.
I hope you have a better Idea of Atonal music (or Serialism)