The experience of music is often compared to the perception of colors. What would it be like to see a completely new color, one you had never seen before? We may never get to experience this, but in the realm of music, we can experience something similar: musical intervals and scales that are completely new to us and outside our normal experience. These intervals fall under what is called microtonalism.
Microtonal music is usually defined as any form of music which uses intervals smaller than the typical 12-step chromatic scale used in most modern Western music.
Microtonal music includes quarter tones and other equally tempered tuning systems, but it also includes other approaches to tuning and composition which are based on small intervals, including compositions using just intonation (exact matches to natural harmonics).