A/F# gives the bass its own instruction: keep the upper chord, but make the lowest note. The slash separates those two jobs.
F# is not in the upper triad; it turns the pitch set into a major sixth chord with its sixth 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 A over F# or A with F# in the bass.

In which keys. Every note of A/F# belongs to C# minor (the ♭VI), D major (the V), E major (the IV), F# minor (the ♭III), A major (the I) and B minor (the ♭VII).
Read A/F# as a minor 7th chord, written this way only in a run of slash chords.
Contains A/F# and adds second.
Contains A/F# and adds second, fourth and major seventh.
Contains A/F# and adds second, sharp eleventh and major seventh.
Adds second, fourth and flat seventh. The flat seventh pushes the sound toward a dominant chord.
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.