Am/C gives the bass its own instruction: keep the upper chord, but make the lowest note. The slash separates those two jobs.
C is already the upper chord’s third, so this is first inversion. 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 C or Am with C in the bass.

In which keys. Every note of Am/C belongs to C major (the vi), D minor (the v), E minor (the iv), F major (the iii), G major (the ii) and A minor (the i).
Contains Am/C and adds fourth and flat seventh.
Contains Am/C and adds fourth, sharp eleventh and flat seventh.
Contains Am/C and adds second, fourth, flat thirteenth and flat seventh.
Contains Am/C and adds second, fourth, sixth and flat seventh.
Contains Am/C and adds flat second, fourth, flat thirteenth 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.