Jazz harmony makes use of intervals and dissonances that are not generally utilised by the 'classical' tradition, except for by modernists such as Stravinsky. Of particular significance is the use of the 'blue notes', so designated because in 'proper' harmony the interval created by them is regarded as unnacceptably dissonant, mistaken, or 'blue'.
The blue notes are largely derived from overlaying 'the blues scale', a simple minor pentatonic scale, across the top of the regular diatonic 7th chords that are generated by harmonising the major or minor scale in four voices.
This practice gives rise to the '7th Sharp Nine Flat Five'-type chords which are so characteristic of the cadence in mainstream jazz. (In fact the 'flattened fifth' is the most recognisable sound in any blues or jazz form of harmony.) Jazz soloists then further accentuate the particular characteristics of these kinds of chords by the use of the 'jazz melodic minor scale' which is derived from the classical melodic minor scale, as well as by the use of the various Greek modes (that is, 7-note diatonic scales commencing from notes other than the root of the major scale) and the Diminished and Whole Tone Scales, to mention a few. (Advanced jazz players have an armory of many subtly different scales and modes to play against any chord.)
Folk, rock and traditional or classical tunes tend to use simple triadic chords (three notes) with the addition of 7th, diminished and augmented intervals where required. Jazz harmonization makes much more use of major and minor seventh chords for even the most basic passages within a song.
Another important characteristic of jazz harmony derives from 'stacking thirds', that is the addition of further notes above the 8th note of the root of the chord, the 8th note being one octave higher than the root. These notes include the 9th, 11th and 13th notes, all of which lend different qualities to the root chord. Depending on the quality of the root chord being played, other intervals may also be added, such as the Sharp 11th.
From the viewpoint of classical theory, many of the chords so formed are in effect 'polychords', that is, notes derived from two different triads played together. Modern jazz has also made extensive use of the interval of the Fourth, which is tonally quite ambiguous and can be used amorphously in a wide range of harmonic contexts (rather like an harmonic form of putty.)
Experienced jazz players also make extensive use of the idea of 'chord substitution' whereby an alternative chord to the one written or implied by the score is played. Depending on the knowledge, skill and 'ear' of the player, there are a practically infinite number of substitutions that may be employed in a tune, giving rise to 'reharmonisation' which will place the melody note of a tune in a completely new harmonic context. Aside from a few standard substitution methods, such as the 'tri-tone substitution', it is difficult to generalise about substitution techniques as a skilled player can draw on a very wide tonal pallette for this purpose.
However despite the enormous range of possibilities inherent in these techniques, it is also important to note that 'it don't mean a thing if it ain't got that swing', as Duke Ellington put it. While 'free jazz' forms and certain schools of (particularly European) abstract modernism have attempted to extend the range of jazz harmony through an almost infinite spectrum, the 'mainstream' schools emphasise a recognizable, through broad, selection of harmonic, melodic and rythmic possibilities, firmly rooted in, among other things, the blues scale, the Gershwin songs ('I Got Rhythm' in particular), the chord voicings of Bill Evans in the Miles Davis 'Kind of Blue' era, and, above all, from the 'canon' of 'jazz standards', many drawn from the 'Great American Songbook', as interpreted by the great jazz artists of yore, and drawn from the Broadway show rather than from the 'Top 40' of the last 30 years.