C/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 puts the upper chord’s fourth in the lowest voice. Hold the upper chord steady and move only the bass. That low line can connect nearby chords without jumping root to root.
Also written C over F or C with F in the bass.

In which keys. Every note of C/F belongs to C major (the I, the tonic), D minor (the ♭VII), F major (the V, the dominant) and A minor (the ♭III).
Read C/F as the root, 5th, 7th and 9th of a major 7th chord, which is how it is usually written instead.
Contains C/F and adds second, sixth and major seventh.
Adds second, sixth 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.