Why is a 2-5-1 progression so often associated with JAZZ?? Is there something inherent in this progression that makes it sound jazzy? Or rather, is it just that jazz people started to use it, experience and re-experience it, and lean on it until it just became part of what jazz is?
To understand the prevalence of the ii-V-I in jazz, there are three good places to look: ragtime, Tin Pan Alley, and blues.
Ragtime was characterized by something called the "Ragtime Progression":
| (V7/V/V/V) | V7/V/V | V7/V | V7 | I |
or more simply:
| III7 | VI7 | II7 | V7 | I |
In the key of C major, this progression is:
| E7 | A7 | D7 | G7 | C |
The first line illustrates the "centripetal" movement around the circle of fourths. In other words, each chord serves a dominant (V) function which leads to the next V chord. But even beyond this, the ragtime progression was interesting to people like Scott Joplin because of the possibility it opened for chromatic movement in the voice-leading, in the melodies, etc. Ragtime composers were so interested in exploring those possibilities that the III7-VI7-II7-V7-I progression became a hallmark of ragtime.
Tin Pan Alley
Then in the early 1900s, songwriters like Cole Porter, Irving Berlin, and George Gershwin came onto the scene. Their music was often referred to as Tin Pan Alley, and their songs were so influential that they form something called the Great American Songbook. These songs are still played with impressive frequency today in jazz. As it turns out, these composers were heavily influenced by ragtime:
The first generation of Tin Pan Alley composers was obsessed with ragtime and its musical and commercial possibilities. They were also interested in repackaging the instrumental ragtime compositions that Scott Joplin and his colleagues had developed as the foundation for their own novelty songs. The decade of Tin Pan Alley's rise, the 1910s, might be usefully marked off by the debut of Irving Berlin's “Alexander Ragtime Band” in 1911 on one end, and George and Ira Gershwin's “The Real American Folk Song (Is a Rag)” in 1918. (p. 3 of http://www.lfpl.org/mylibraryu/pdf/Session_Two_Materials.pdf)
Given the massive influence of these Tin Pan Alley composers and the endurance of their songs, the II-V-I progression would forever become an integral part of jazz harmony. But in part because Tin Pan Alley composers often wrote for musicals, their music followed specific forms like AABA, AAB, AB, etc. which each letter (A and B) marking different sections. For variety, these composers used different sections to modulate to different keys. This had two effects: (1) the prevalence of the ii-V-I grew because it's perhaps the easiest/most natural way to modulate, and (2) the ii-V-I became preferred over the II-V-I, partly because the ii-V-I was entirely diatonic and helped establish the new tonal center more quickly.
Blues (and Bebop)
There's a second unique lineage that also explains the prevalence of the ii-V-I. The core 12-bar structure that evolved from blues is something like this (in C):
| C7 | C7 | C7 | C7 |
| F7 | F7 | C7 | C7 |
| G7 | F7 | C7 | C7 |
As jazz musicians began using the blues progression, many sought ways to increase its complexity while preserving its original structure. This was especially true of the bebop musicians. People like Charlie Parker were known for adding chords into a song as a way to enable more complex melodies and improvisations. The easiest ways to add in extra chords are: (1) add the V7 chord in front of any "orphaned" I chords, and (2) add the ii chord in front of any "orphaned" V7 chords. The result is an explosion of ii-V and ii-V-I progressions in bebop music. For example, many bebop musicians have taken the blues structure above and modified it this way:
| C7 | Bø E7alt | Amin D7 | Gmin C7 |
| F7 | F7 | C7 | Emin A7 |
| Dmin | G7 | C7 A7 | Dmin G7 |
All we've done here is add additional V7 chords and ii chords. This procedure of adding ii-V7 chords quickly spilled over into many other songs that didn't have the blues structure. And this simply increased the prevalence of the ii-V-I even more.
One reason the ii V7 I progression is associated with jazz is that jazz developed alongside the music of the Great American Songbook. The tunes in the Great American Songbook are written using ii V7 I progressions and move through different keys.
Jazz musicians started improvising over these tunes, and then began writing new heads over the existing progressions. For example, many jazz tunes (such as Charlie Parker's "Anthropology" and Sonny Rollins' Oleo) are written over the changes to "I got Rhythm" and the Miles Davis tune "Dig" is written over the changes to "Sweet Georgia Brown." There are many such examples. They also took the ii V7 I concept and re-wrote the blues progression to make use of it. It probably made things more interesting for the musicians and the audience.
As a consequence, many jazz tunes written in the 30's 40's and 50's utilized the ii V7 I progression.
Notably, many modal tunes do not utilize the ii V7 I progression. Miles Davis' tune "So What" is a good example of this. So at a certain point jazz musicians began to move away from the ii V7 I progression. The resulting sound is arguably more open and free, but is also more challenging for players to create truly interesting solos over modal tunes.
At any rate, the ii V7 I progression continues to be a very important element of jazz.
What @Tim answered.
In addition, and by no means a contradiction:
ii-V7-I follows the circle of 5ths : F-Bb-Eb is ii-V7-I, etc. So ii-V7-I facilitates logical, systematic, fluid movement in melodies and improvisation, as you work through circle of 5ths. (Sometimes it's referred to as the Circle of 4ths, particularly when moving up the scale counter-clockwise - the flat direction.)
Listen to how jazz soloists so often build their solos in ii-V7-I tunes - you'll hear that an important component in the musical motion comes from climbing up and down through the notes connecting the scales of the Circle, lingering a bit and exploring the new scale/chord, and then moving on to the next one. That makes the music flow in an engaging and pleasing manner.
At the root of it all in terms of jazz's history is that the jazz repertoire in its early days was comprised to a large extent of popular tunes - tin pan ally, vaudeville/show tunes, etc - generally known today as
Standards. Those tunes were used as the framework for jazz improvisation, and those tunes made ample use of the ii-V7-I progression, for the reasons stated - it sounds good, and works well theoretically speaking. Thus ii-V7-I became integral to the jazz lexicon.
That's rather an opinion-based question! It's not specifically associated with jazz. A whole lot of popular music makes great use of ii, V, I or an even more expanded 'cycle of 5ths' sequence. (And the rest of it is based on the Blues ;-)
A whole lot of later-period jazz ignores 'functional' harmony altogether and noodles around modally on progressions like Am9, Gm9 repeated continually.
Instruction books on jazz improvisation do tend to overdose on ii, V, I sequences in all the keys, maybe because it IS such a common pattern in mainstream songs.