Modes are scales (like do, re, mi...) that apply only to the melody string on a dulcimer, and do not include any plus frets. The do, re, mi fa, sol, la, ti, do scale that we learned in grade school is the Ionian Mode, and starts on the 3rd fret of a diatonic fretboard.
Notice that these scales or Modes are independent of any actual notes like a C or G or A...
Each Mode starts at a different fret (as the chart shows), and certain notes in the Modes are sharper or flatter than the "normal" Ionian notes.
For example the Mixolydian Mode has the famous "flatted 7th" note -- do, re, mi, fa, sol, la, ti-flat, do. The reason modern dulcimers have the 6+ fret is so that you can have both the "ti-flat" of the Mixolydian Mode and the "regular ti" of the Ionian Mode on the Mode/Melody string at the same time. On modern dulcimer, when you start a scale at the Open note (Mixolydian Mode) the ti-flat note is fret 7, the 'regular ti' is the 6+ fret.
Modal Tunings are what we mean when we think of Dorian or DAA or 1-5-8 or whatever. They are a combination of the Mode string plus two drones.