F/D# gives the bass its own instruction: keep the upper chord, but make the lowest note. The slash separates those two jobs.
D# is not in the upper triad; it completes a dominant seventh chord with its flat seventh lowest. Hold the upper chord steady and move only the bass. That low line can connect nearby chords without jumping root to root.
Also written F over D# or F with D# in the bass.

In which keys. Every note of F/D# belongs to G minor (the ♭VII) and Bb major (the V, the dominant).
Read F/D# as a lydian chord: the triad supplies the ♯11.
Contains F/D# and adds ninth, eleventh and thirteenth.
Adds ninth, sharp eleventh and thirteenth, for a dominant sharp-eleventh color.
Contains F/D# and adds flat ninth, sharp ninth, sharp eleventh and thirteenth.
The common shapes and the derived voicings come from different places, and both are explained in how this site is built.
Shows each chord the way this style would voice it, and marks which of its alternatives belong to the idiom.