In jazz music, the ability to improvise with soul, creativity, and individuality, while still remaining organically connected to the sound being put out by the band as a whole.
The ability to swing is what distinguishes a good jazz player from a bad one. Lack of soul (Kenny G) or showboating (Jaco Pastorius toward the end of his career) are antithetical to swing.
Normally used in the context of acoustic jazz, but equally applicable to fusion, jazz-funk, etc. Not be confused with "swing" as a genre. (While a good Swing musician must know how to "swing," one can swing in just about any genre involving improvised music).
Man, that sax player sure knows how to swing.
I wasn't feeling the band last night. They kept trying, but they just couldn't swing.