The classic example is Bb/D (xx0331) - (Used often by Brian May from queen). Tonally, it's a Bb chord, i.e. it sounds
like a Bb, bt the lowest note (as Chris said the bass note) is a D.
Something like Dm/C would look like this (x3x321). The D string is muted as, in most cases, it would sound too much like a Dm7 chord.
Hopefully this is making at least some sense...There's no real secret behind it, it's just one of those things that isn't often explained and therefore quite a few people just don't know about it...
Don't worry too much about me, ignore me long enough and I'll go away.