About D/C — uncommon voicings
D/C is C, D, F# and A — with C put in the bass.
It sounds like the major triad it is, resting on a different floor. If C is already one of the chord’s notes this is an inversion, and if it is not it is a new colour under an unchanged chord; either way the reason to reach for it is a bass line that walks by step where root-position chords would leap.
Inversion or pedal — the harmony is unchanged; only the bass moves.
Also written D over C or D with C in the bass.
What these less common shapes are
How to use D/C
In which keys. Every note of D/C belongs to E minor (the ♭VII) and G major (the V, the dominant).
What to play over D/C
Read D/C as a lydian chord: the triad supplies the ♯11, which puts it in C lydian. It also sits inside D mixolydian, D lydian dominant and D half-whole diminished.
Each of those is a page: the scale across the neck, and the chords it harmonises into — which is where D/C sits in it. Every scale.
Chords near D/C
Change one note of D/C and you have (drop the flat seventh), (the flat seventh becomes the major seventh), (the major third becomes the minor third), (the flat seventh becomes the ninth), (the flat seventh becomes the sixth) or (the major third becomes the fourth).
D/C is a slash chord — a triad over a named bass note. Slash chords on every root.
The common shapes and the derived voicings come from different places, and both are explained in how this site is built.