Jazz guitar can be difficult to master because improvisation (making up music on the spot) is such an important part of the style. Normally, making up the music is the job of the composer. But in jazz, the performers are (usually) expected to improvise.
Jazz is a form of music that instrumentalists created when they began taking liberties with existing song forms, improvising off composed melodies, and varying harmonic structures. Guitarists followed the early efforts of other instrumentalists like the great trumpeter Louis Armstrong, who was one of the first early masters of melodic improvisation.
Jazz guitar differs from rock and blues guitar most significantly in the following ways:
Jazz guitar doesn't use distortion, favoring a softer, mellower tone.
Jazz melodies are more harmonically sophisticated, observing more closely the chord constructions — which are themselves more complex.
Jazz lines often employ more skips (musical distances of more than a step — for example, A to C) than rock or blues lines do.
A jazz guitarist's approach to chords is deeper than a rock or blues player's. In rock and blues, guitarists typically use one scale to play over all the chords, but in jazz, they may use many scales. They also must be aware of the notes that make up each chord, as arpeggiating, or playing chord tones in succession, is a hallmark of the jazz sound.
Most of the music you hear — pop, rock, blues, folk, and classical (especially classical music from the 17th and 18th centuries, like that of Bach and Mozart) — relies on traditional harmony (basic chords and progressions). But jazz harmony uses what most people call (big surprise) jazz chords. Jazz chords often contain more notes than basic chords, or sometimes they can have the same number of notes as basic chords, but one or more of their notes is chromatically altered (raised or lowered a half step).
Extended chords
Simple major and minor chords are made up of only three notes — the 1st, 3rd, and 5th degrees of the major or minor scale whose starting note is the same as the chord's root. These chords are called triads (three notes). Seventh chords are made up of four notes — the 1st, 3rd, 5th, and 7th degrees of the chord's namesake scale.
In jazz, you find chords made up of five or more notes. By continuing to take every other scale degree, you can go beyond the 7th to create 9th chords (using the 1st, 3rd, 5th, 7th, and 9th degrees), 11th chords (1st, 3rd, 5th, 7th, 9th, and 11th), and 13th chords (1st, 3rd, 5th, 7th, 9th, 11th, and 13th). These chords that include notes beyond the 7th are called extended chords.
Usually, not all the members of an extended chord are actually played. For example, in a 13th chord, you may play only four or five of the seven notes, so it's possible to play a 13th chord by using only four strings.
Altered chords
Jazz chords often contain notes that are altered (raised or lowered a half step). These alterations produce all sorts of funny-sounding chord names, like C7@9, B@13#11, and G7#5. And each of these jazz chords — and dozens of them exist — has a unique sound.
In playing jazz versions of popular songs, altered chords are usually substituted for more traditional chords — but knowing which chord to substitute, and when, is no easy feat and requires the skill of an accomplished jazz musician.