Tonal music is something that revolves around a certain tone or a key. Most of the Western music and music of popular genre are tonal music and very pleasing to our ears. All the discussions we have had about keys and modes basically revolve around a tonal centre which we also refer to as the home. It starts and ends on that tonal centre and therefore pleasing to our ears. Atonal music on the other hand is something that doesn’t follow any structure and doesn’t have any specific tonal centre. A good analogy for atonal music would be an abstract painting. How one is going to perceive an abstract paining is entire subjective and different people may give different meanings to it. Same is true for a piece of atonal music. Following is an example that demonstrates the characteristics of atonal music. It may sound completely absonant but if you listen to the music several times, it starts to make some musical sense.
Atonal music goes completely against the classical norm of music composition and a relatively new way of composing music. The idea originated in 1908. There are several 19th and 20th century composers (i.e. Alexander Scriabin, Béla Bartók etc.) who are known for their atonal compositions. Atonal music is also labelled as modern classical music by some people. So if you pick up the guitar and play some completely random notes, you may actually end up making some form of atonal music!