It means that you play a D chord with a F# at the base (first string you hit) instead of the root (D).
Major chords are build with the root, the 3rd, and the 5th of the octave/major scale (being the D major scale here).
So a D chord is build with D as the root, F# as the major 3rd and A for the 5th. F# is the 3rd of the D chord and when you play the 3rd first, instead of the "normal" way of playing (the root first), its called a first inversion chord.
You can also build a 2nd inversion chord by starting with the 5th first.
So D over F#, or written this way D/F# is simply a D major chord first inversion to create a different voicing.