Zero Frets on Guitars

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On an instrument with a "zero fret", the traditional slotted nut still controls the strings spacing but the zero fret (which is actually fret wire) now controls the height of the strings above the frets. The strings will lay directly on the zero fret and string height will be adjusted by filing or replacing the zero fret instead of cutting the slots in the nut deeper.
 

On instruments with zero frets the scale length is measured from the zero fret to the saddle as opposed to the nut to saddle.

 
Technically speaking, when you play a C chord, for example, some notes of the chord are fretted others are open.  This means the fretted notes rest upon a fret when resonating and the open notes rest upon a bone/plastic nut. When fretting that chord on an instrument with a zero fret, all strings, even the open ones rest upon a fret.
Now, take that and run with it.

Theoretically you could argue that this sounds like a very good thing, but in the real world you'll find a variety of opinions.