Slash-Chord. On the left side of the slash you have a chord, on the right side the lowest tone (bass). The C/G (say „C over G”) is a C major chord with a low G played in the bass.
The C major chord consists of the tones C (1), E (3) and G (5). The bass on the right of the slash can indeed, but does not necessarily be part of the chord on the left. In this case the bass is the C major chords 5th.
This major chord with its 5th in the bass is often used when several major chords are played over the same bass note (pedal point or pedal tone, pedal note, organ point, or just pedal). Chord progression example with G pedal: GC/GD/GC/G (= Ist, IVth and Vth scale degree of the G major scale with Ist degree in the bass) C/GF/GC/GG (= Ist, IVth and Vth scale degree of the C major scale with Vth degree in the bass)
In a different common progression the C/G chord appears as Am7/G: Am7 Am7/GD/F#F
In this case the chord is written Am7/G, because the guitar player will find out easier that he just has to move down the bass from A to G. In doing so, he will normally abandon the A, so that the chord actually becomes a C/G. But for the sake of better readability in this progression the writing Am7/G is preferable.