The harmonic minor scale is commonly used by jazz musicians. It has a darker "Arabic" sound because of the lowered third and sixth degrees (when compared to major). It has been used, much like the major scale, in traditional and classical music for centuries.
Please refer to the minor scale article in Wikipedia if needed.
On the guitar it can be played in the open position :
Notice the leap created by the largest interval between two consecutive notes :

Also, it is highly common for jazz guitarists to sing along to their playing. Try it! It helps to reinforce the link between your fingers (what you play) and you ears (what you hear inside.)
The harmonic minor scale is built of seven notes that are laid out using intervals. The half-step interval is one fret away, the whole step is two frets away and the minor third is three frets. The minor third can also be qualified as "three half-steps" or "a whole step and a half".
The formula for the construction is :
W H W W H -3 H
(W= whole-step, H=half-step, -3=minor third)
The numeric formula is : 1 2 b3 4 5 b6 7
(it means that in comparison to the major scale, the harmonic minor scale has a b3 and b6)
The lowered third degree gives the "sadness" in the sound. The b6 on the other hand gives "brightness" because it creates a leap to the seventh degree. It easy to "see" when played on a single string.
It is a very good way to "compartmentalize" the neck. You will get familiar with the whole instrument dealing with it chunk by chunk.

Jazz Guitar Chords and Jazz Harmony