In a jazz context, Dom's answer is perfect.
However, I also want to say that this chord has become extremely common in pop/rock piano-driven music, for example Elton John, Billy Joel, and Stevie Wonder. And especially in musical theater of this style, for example Jason Robert Brown, Stephen Schwartz, and the "straight to off-off-Broadway" duos like Kerrigan-Lowdermilk and Salzman-Cunningham.
In this context, when you have a chord like F/G resolving to C, I think the simplest explanation is that it's primarily an F chord (and thus you have a plagal cadence), with a 5->1 in the bass that strengthens it a bit by adding a bit of dissonance to be resolved, plus outlining an authentic cadence. If you think of it this way, then F/G is really the end of the story. It's an F chord, with a G in the bass.