First introduced in 1960 as the Deluxe Model, it borrowed design elements from the Jazzmaster guitar.
It was renamed the Jazz Bass as Fender felt that its redesigned neck—narrower and more rounded than that of the Precision Bass—would appeal more to jazz musicians.

The Jazz Bass has two single coil pickups with two pole pieces per string.
As well as having a slightly different, less symmetrical and more contoured body shape (known in Fender advertising as the "Offset Waist Contour" body), the Jazz Bass neck is noticeably narrower at the nut than that of the Fender Precision Bass.
While the Precision Bass was originally styled similarly to the Telecaster guitar, the Jazz Bass' styling was inspired by the Jazzmaster guitar, with which the Jazz shared its offset body and sculpted edges that differentiate it from other slab-style bass bodies.

The original intention of the instrument was to appeal to upright bass players.
The original Jazz Bass had two stacked knob pots with volume and tone control for each pickup.
Original instruments with this stacked configuration are highly valued in the vintage guitar market.
In late 1961 it received three control knobs: two controlling the volume of each pickup and one the overall tone.
Despite this new feature, many stacked knob models were made until about 1962.
Another feature the initial models had were the "Spring Felt Mutes", which were present on basses from 1960 until 1962.
The purpose of those mutes was to dampen the overtones and the sustain, and were screwed in place between the bridge and aft pickup.
Those felt mutes were not a tremendous success, and were replaced by a cheaper, more simple foam mute glued underneath the bridge cover as was used by the Precision Bass from 1963 onwards.
Over the following years as the use of mutes gradually declined both the Precision and Jazz Bass models eventually began to be produced without bridge/tailpiece covers.

A number of cosmetic changes were made to the instrument when CBS purchased the Fender companies in 1965.
During 1965/66 the Jazz Bass received bound rosewood fingerboards with pearloid dot position inlays (which replaced the older "clay"-style of the early 1960s) and oval-shaped tuning machines.
Block-shaped fingerboard inlays and an optional maple fingerboard were introduced after 1966/67.
At first necks with rosewood fretboards received pearloid blocks/binding and maple fretboard necks received black.
Fender switched to pearloid blocks/binding on all necks in mid-to-late 1973.
Fender also switched to the three-bolt neck "micro-tilt adjustable" neck and the "bullet" truss rod in mid-to-late 1974 before reverting to the more standard four-bolt neck fixing and dot-shaped fretboard markers in 1983.
White pickup covers and a pickguard/control plate were introduced the same year.
In 1986 Fender introduced the Japanese-made Fender Performer Bass, also with micro-tilt neck, designed by John Page and intended to be an Elite version of the Jazz Bass; however, the radical styling was not popular and production ceased the same year.

Two other changes that were more important to the tone of the instrument also occurred in the early 1970's.
From 1960 until around 1971, the two pickups on the Jazz Bass were 3.6" apart.
The bridge pickup was then moved .4" closer to the bridge, creating a spacing of 4".
 According to Fender itself, this change happened in 1972.
However, there are unquestionably many examples of Jazz basses from 1971 that use the 4" pickup spacing, and there are Jazz basses made in 1972 that use the older 3.6" spacing.
Many players believe that this change contributed to a somewhat brighter tone from the bridge pickup.
Around the same time, Fender began using ash for most of the instrument bodies.
 Prior to the early 70's, nearly all Jazz basses had bodies made of alder, except for those that were finished in a clear or("natural") finish - for those basses alder was nearly always the wood of choice.
Ash is generally thought produce a somewhat brighter (and correspondingly less warm) tone than alder.
  
American Standard Jazz
First introduced in 1960 as the Deluxe Model, it borrowed design elements from the Jazzmaster guitar. It was renamed the Jazz Bass as Fender felt that its redesigned neck—narrower and more rounded than that of the Precision Bass—would appeal more to jazz musicians.
The Jazz Bass has two single coil pickups with two pole pieces per string. As well as having a slightly different, less symmetrical and more contoured body shape (known in Fender advertising as the "Offset Waist Contour" body), the Jazz Bass neck is noticeably narrower at the nut than that of the Fender Precision Bass. While the Precision Bass was originally styled similarly to the Telecaster guitar, the Jazz Bass' styling was inspired by the Jazzmaster guitar, with which the Jazz shared its offset body and sculpted edges that differentiate it from other slab-style bass bodies.
The original intention of the instrument was to appeal to upright bass players. The original Jazz Bass had two stacked knob pots with volume and tone control for each pickup. Original instruments with this stacked configuration are highly valued in the vintage guitar market. In late 1961 it received three control knobs: two controlling the volume of each pickup and one the overall tone. Despite this new feature, many stacked knob models were made until about 1962. Another feature the initial models had were the "Spring Felt Mutes", which were present on basses from 1960 until 1962. The purpose of those mutes was to dampen the overtones and the sustain, and were screwed in place between the bridge and aft pickup. Those felt mutes were not a tremendous success, and were replaced by a cheaper, more simple foam mute glued underneath the bridge cover as was used by the Precision Bass from 1963 onwards. Over the following years as the use of mutes gradually declined both the Precision and Jazz Bass models eventually began to be produced without bridge/tailpiece covers.
A number of cosmetic changes were made to the instrument when CBS purchased the Fender companies in 1965. During 1965/66 the Jazz Bass received bound rosewood fingerboards with pearloid dot position inlays (which replaced the older "clay"-style of the early 1960s) and oval-shaped tuning machines. Block-shaped fingerboard inlays and an optional maple fingerboard were introduced after 1966/67. At first necks with rosewood fretboards received pearloid blocks/binding and maple fretboard necks received black. Fender switched to pearloid blocks/binding on all necks in mid-to-late 1973. Fender also switched to the three-bolt neck "micro-tilt adjustable" neck and the "bullet" truss rod in mid-to-late 1974 before reverting to the more standard four-bolt neck fixing and dot-shaped fretboard markers in 1983. White pickup covers and a pickguard/control plate were introduced the same year. In 1986 Fender introduced the Japanese-made Fender Performer Bass, also with micro-tilt neck, designed by John Page and intended to be an Elite version of the Jazz Bass; however, the radical styling was not popular and production ceased the same year.
Two other changes that were more important to the tone of the instrument also occurred in the early 1970's. From 1960 until around 1971, the two pickups on the Jazz Bass were 3.6" apart. The bridge pickup was then moved .4" closer to the bridge, creating a spacing of 4".  According to Fender itself, this change happened in 1972. However, there are unquestionably many examples of Jazz basses from 1971 that use the 4" pickup spacing, and there are Jazz basses made in 1972 that use the older 3.6" spacing. Many players believe that this change contributed to a somewhat brighter tone from the bridge pickup. Around the same time, Fender began using ash for most of the instrument bodies.  Prior to the early 70's, nearly all Jazz basses had bodies made of alder, except for those that were finished in a clear or("natural") finish - for those basses alder was nearly always the wood of choice. Ash is generally thought produce a somewhat brighter (and correspondingly less warm) tone than alder.   
American Standard Jazz
Fender Jazz Bass