What are C/G and G7/D chords? I have been unable to find out. They are used in Heart of Stone by the Stones.
Do you know what C and G7 chords are? Play those chords but with the lowest note being the note after the slash. So for G7/D, play G, B, D, F but make sure the lowest note is D. On the guitar, you may need to change come of the other frets to make sure you have G, B, F represented while keeping D in the bass. If you have a separate bassist then the guitar player doesn't need to change anything, just the bass player.
Generally, if the note after the slash is the third of the chord (C/E; G7/B; etc.) you can just play the chord without changing the bass and it'll sound pretty similar. If the note after the slash is the fifth of the chord (C/G; G7/D) then it's pretty important since it alters the sound of the chord substantially.
I like Michael Cuthbert's answer, but I think the question could be answered a little bit more concisely and would like to clarify a few things.
I read these chords like "C bass G" and "G7 bass D". They're related to inversions in that they change the sound of the chord without changing its tonality; because they directly influence the way the chord sequence sounds, they are often used to specify bass lines (giving "Never Grow Up" a descending feel during the verses).
What Michael said about the guitarist/bassist thing is definitely worth heeding too!
On guitar in particular, it's about VOICING. Because there are several different ways to voice any chord,the slash thing gives, as said above, the lowest note played .This does not necessarily make the chord a 'root', '1st' or '2nd' inversion.It just changes the voicing of the chord.
Often it allows the sequence to have the 'bass' line moving up or down a scale, or chromatically, which wouldn't happen if the chords were just stated normally.As above, also, if there is a bass player, he'll be expected to do the slash bits rather than, say, play root notes.
It's also possible to have the slash bit as a note not in the chord. So C/B would be a Cmaj. with a B bass. Could be called Cmaj7, but the composer felt that the main chord needed to be pure Cmaj.