"how this morphed into the 'atonal' style, and what the significance of 'atonal' is"

Atonal simply means without a tonal center. Tonal music is based on the idea that one "tone" is more important than the rest. This "tone" provides the point of departure and conclusion for a piece of tonal music, thus explaining our satisfaction once we reach the end and achieve "tonal resolution." With this expectation comes the power to deter, impede, and "deceive." In most tonal pieces, drama arises from establishing a key, moving away from it, and returning. This is the basic idea of sonata form, a design which forms the basis of most of Beethoven's major works.

As time passed, the methods for creating this tonal drama intensified (i.e. composers had to move farther and farther away from the tonal center). This was necessary because the traditional key relationship became "old hat" and unexciting as the 19th century progressed. As a result, the tonal system started to break down. You can hear this most clearly in the works of Wagner, Strauss, and Mahler. The sense of one "tone" or key as most important is vague and passing. This helps to create the sense of "longing" or "desire" in their music–as the music truly avoids closure or resolution.

By the turn of the century, composers like Schoenberg, Webern, and Berg felt that the expressive potental of tonality had been used up and over explored. They turned to explicitly creating atonal works–works without any sense of a center or need for resolution. All of these composers were also influenced by Brahms' technique of varied repetition, so there is little direct repeptition in their music–which can make it diffucult to grasp on one hearing. Their music might also be called "expressionist"–depicting intense, dark, and terrific psychological states.

Some people have compared tonality and its demise to social structures in European society. Traditional tonality is like feudal system–there is a ruler (tonic–the "tone"), to which all other tones are subservient, etc. The breakdown of tonality follows a similiar breakdown of "centrality" in European society.