Jazz music SOUNDS and APPEARS very complicated. However, 75% of fundamental Jazz undertanding AND improv is guided by a single theoretical rule:
ii V I
Say "Two Five One" to any Jazz Musician and their face will light up, assuming you 'get it'.
How's it work? Well ...
Western Music includes 12 UNIQUE NOTES TOTAL.
From these 12 notes, a fixed regular pattern is used to select 7 notes. These subsets of 7 notes are called "Major Scales" (or "Keys"). Thus, there are 12 UNIQUE MAJOR SCALES.
However, the real 'game' of Jazz depends NOT on notes, but chords.
Within each Major Scale, fixed regular patterns are used to select CHORDS (two or more notes played at the same time). Although Jazz is embelished by complex chords (ex: B7b9, say : "B seven flat nine"), the understanding and recognition of Jazz patterns depends only on MAJOR and MINOR CHORDS.
Jazz Musicians represent the CHORDS OF EACH UNIQUE MAJOR SCALE in a 2-dimensional Roman numeral notation as follows:
I ii iii IV V vi vii*
Dimension 1: The NUMERIC VALUE of the roman numeral == the number (1-7) of the note from the major scale upon which the chord was built.
Dimension 2: The CAPITALIZATION of the roman numeral indicates major or minor.
Upper case == major.
Lower case == minor, where * == insignificant variation.
The game of Jazz is to use ii V I CHORD SEQUENCES from different major scales and to combine them together. Thus a typical Jazz Song would go as follows (for abstract Major Keys X, Y, and Z):
Start in Major Key X
==> Play ii V I
SWITCH to Major Key Y
==> Play ii V I
SWITCH to Major Key Z
Play ii V I
Although there are guidelines for how, when, and why to SWITCH, there are no fixed rules, which is where theory ends and application begins.