Traditionally, Western music has utilized what is called tertiary harmony. It's called "tertiary" because we create triads, or chords, by taking a note and stacking two intervals of a third on top. Creating a triad off of all 7 notes of our chosen scale gives us our diatonic chords of that scale.
Extending off of this idea, a system of quartal harmony would have us create our chords by stacking fourths on top of each note in the scale rather than thirds. The resulting harmonies sound quite different and are a bit "alien" to our ears. Quintal harmony works the same way, but instead of stacking fourths, you stack fifths instead. Both systems are somewhat similar to each other, since both 4ths and 5ths can either be perfect, augmented, or diminished.
Composers will use quartal or quintal harmony when they want to create a ambiguous, open, or ominous sound.