For an explanation of Just Intonation and what all these ratios are, go here.
Ben Johnston's notation system for Just Intonation aims to produce something that looks similar enough to conventional notation that the average musician can get their head round it without going too batshit insane.
The basic notation is the C major scale. The notated triad C-E-G, F-A-C and G-B-D all represent just major triads, in ratio 4:5:6 (e.g. if C is 1/1 [=4/4], then E is 5/4 and G is 3/2 [=6/4])
One consequence of this is that the interval D-A isn't a perfect fifth, it's actually 40/27. This arises because of the syntonic comma, the difference between a just-tuned major third and a Pythagorean major third (i.e. one derived through the so-called "cycle of fifths"). This small interval is a ratio of 81/80, and that's notated by a + or a - (line a above)
(Another consequence of this is that if you tune to "A", as is standard practice, the open strings of a cello are therfore not A, D, G, C but A, D-, G-, C-. I experimented for a while with a version of this notation in which a key signature of one sharp indicates that the uninflected notes of the G major scale are the 4-5-6 triads instead of the C major scale. This meant that for a cello the open strings would be notated as naturals (although a violin's E string would have to be E+). I decided this was potentially too confusing and I've now reverted to Johnston's original plan.)
b) sharps and flats raise or lower a note by the ration 25/24, which is the difference between a just major and minor third (5/4 and 6/5)
c) the 7 and upside-down-7 signs raise or lower by 36/35, which is the difference between a just minor ninth (9/5) and the "flat" seventh partial in the harmonic series (in C major, Bb7)
d) the up and down arrow raises or lowers by 33/32, which is the difference between a perfect fourth and the "sharp" fourth that is the eleventh partial (in C, F^)
e) the "13" and upside-down "13" raise or lower by 65/64, which is, you guessed it, the difference between a minor 6th and the "sharp" minor 6th of the 13th partial (Ab13).
You can go on ad infinitum, adding a new inflection for every prime number, but that's quite enough to be going on with. f), g) and h) show a few examples of how these accidentals may be combined to form compound signs, except the +/- of 81/80, which is always separate and always closest to the note.