Diatonic chords are chords that are made up strictly of diatonic notes. There are seven diatonic notes per key; each of these notes is the 'root' of a diatonic chord. Diatonic chords can be triads, 'seventh' chords, or 'extended' chords.
Commonly, all of these diatonic chords are played as simple triads, except for the 'V' chord (the chord whose root is the fifth note of the scale). The V is commonly played as a 'seventh' chord (containing the first, third, fifth, and seventh notes as counted on the scale from the root of the chord), and is notated in the Nashville system as V7. The 'seven' chord (vii), (not to be confused with the V7), based on the seventh note of the scale, could also be played as a 'half-diminished-seventh" (notated as ø or ø7), and that's how we'll use them here.