Am/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 minor 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 Am over F# or Am with F# in the bass.

In which keys. Every note of Am/F# belongs to E minor (the iv, the subdominant) and G major (the ii, the supertonic).
Contains Am/F# and adds second, fourth and flat seventh.
Contains Am/F# and adds second, fourth and major seventh.
Contains Am/F# and adds flat second, major third, sharp eleventh and flat seventh.
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.