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By stacking 4ths in C major, the following quadads result:
(I used C major for simplicity, but this can obviously be transposed to any key)
They are easy to play, guitar-friendly chords. Most (if not all) of them sound really cool and I love to play them. Where can I use them in the context of an existing harmony?
You can either use them as "sounds" in a more impressionistic way if you play a modal piece (just like on the whole "Kind of Blue" album mentioned in Shevliaskovic's answer). But if you want to play them in "the context of an existing harmony", assuming you mean standard harmony based on thirds, then you can try to figure out how those quartal chords are subsets of standard chords. Here are some examples to get you started:
D G C F can be used as a voicing of Dm7 (not 5th, added 11).
E A D G can be used as C6/9 (no root)
F B E A can be used as G13 (no root)
You see, all those chords are a bit ambiguous if interpreted as voicings of standard chords, but that's exactly why they sound so refreshing. If you play around a bit you'll find many more possibilities like the ones I've mentioned above.
A good example is used by Bill Evans on So What:
Bill used the kind of chords you mentioned, while adding a third at the end.
The theorists have categorized these kind of chords as 'So what chords'
There are many modal jazz songs that use these kind of voicings; like 'Impressions' by John Coltrane.
So, you can feel free to use them on modal jazz
while triad-based chords imply specific harmonies, quartal chords are more generic and basically don't imply specific harmony in the key you're in, which means you can use them pretty much anyplace and in lieu of regular chord progressions. Quartals are more vague so they fit in a wide variety of situations.
Quartals can also be thought of as voicings for extended chords (9ths, 11ths, 13ths) and so therefore can be used as ways of voicing triads when you want to use triads with their tensions (extended notes).
A good explanation of this is in the short booklet: Jazz Piano Comping: intermediate, by Tom Anderson.
A good real-life illustration of how these chords are used on piano is found in any of the Jamey Aebersold "Piano Comping" and "Transcribed Piano Voicings" volumes. These are the transcriptions of the pianists who are comping along Aersold's play-along recordings. When you look at what they're doing, you see that they use quartals a lot and they use them very freely regardless of what the written set of chord changes is supposed to be.
