Dear Pies:
There is only one way to describe or explain music: listening to it. If you have to introduce someone to non-tonal music, and this person is acquainted with tonal one, show examples of both tonal and non-tonal works and compare the various parameters: sound, harmony, melody, phrases. Perhaps the most important characteristics of atonal music are easy to recognize, however a deep exploration and comprehension demands a lot of study and knowledge, and - paramount - a very well developed ear.
By the way, the musical capacities of the listener are important always, and just through them someone is able to understand and appreciate music in its complete scope.
Best wishes!