What is this called?

Chords such as the "D / F#" you cite as an example are called "slash chords."

Usually when one plays chords on the guitar, the chord's root note (the note that shares the same note name as the chord - E is the root of E or Em, A is the root note of A or Am, F# is the root note of F# or F#m, etc.,) is the lowest note played. But it doesn't always have to be.

Strum an A chord. Most people will strum this from the A string down. It certainly sounds cleaner than if you strum it from the low E string down. Even though the E note is part of the A chord, on the guitar the A sounds less muddy if you strum the A note as the chord's lowest note.

But you can certainly strum all six strings to make an A chord. Technically, this would be "A / E" - meaning it's an A chord but you're now using E as your lowest bass note.

You want to look at slash chords like this:

Chord / Bass note other than the root note of the chord

So your "D / F#" example is a D major chord with the F# note (second fret of the low E string) as its bass note. The way most people play this is to start with a regular D chord and then wrap the thumb of your fretting hand around the neck so that you can rest part of it on the low E string at the second fret. You don't need all that much of your thumb to do so.

Hope this helps. I'll try to write up a "Guitar Noise Quick Guide" lesson on this over the weekend with chord chart examples of the ones guitarists often run across.

Peace

One quick addition to David's fine explanation - when you're pronouncing D/F# you say "D over F sharp". That tells you immediately what it is: a D chord played over an F# bass note.