Atonality and Serialism
Up to this point, we've mostly been concerned with scales and compositions based around a particular tonic note. Even where the key modulates, we have modulated to and from keys with a tonic note - at any point in time, the composition has some tonal center, although it is not the same note throughout the composition. Now we will cover scales, and particularly compositions, without a clear tonic note. This is called Atonality.
Free Atonality
Atonality is not a strictly-defined term. It is a blanket label that is often applied to any music that is sufficiently difficult to understand in terms of traditional tonal analysis methods. This may mean that the piece contains many chromatic notes outside of the perceived key, or that there are many non-diatonic chords beyond simple tonicizations via secondary chords. It can also apply to pieces that only use diatonic notes within some key, but whose melodies do not tend towards any suspected tonic note, or whose chords do not fit into the harmonic functions that diatonic chords normally represent.
As you can imagine, atonality basically takes a wrecking ball to everything we've learned about music theory. Early atonal music around the early 1900's was pretty chaotic in this respect, known as free atonality, with the term "atonal" often used as an insult implying its composers didn't understand tonality.
Many speculate that atonality is not possible, in that the human psyche strives to find tonality in even seemingly randomly created music. Atonal music was later refined and structured through Serialism, in order to be as truly "atonal" as possible.
Serialism
Serialism is a compositional technique that uses rows of some musical element, such as rhythm or pitch. A row is a set of different elements arranged so that each element is played in sequence with each member element represented. The exact sequence chosen is up to the composer. When talking about pitch, this means that every note in the pitch class is played in every row.
Even further, no note is played more than once per row. The most common pitch row is the Twelve-Tone Row, which uses every note from the chromatic scale. Although notes can be different durations, have different dynamics, and fall on different beats, every note is represented, forcing the composition away from any tonality.
While serial music is not necessarily atonal, it is highly associated with it. For example, if our row consists of notes strictly from C Major, the piece could have a clear tonic note of C. Yet serialism typically uses a twelve-tone row, abandoning any attempt at tonality.