This little diagram gives you an easy pattern that'll help you quickly remember the scale for any of the modes listed below.
First, you need to know the major (Ionian) scale patterns. What the chart gives you is a way to find out the major scale that contains the same notes as the mode you want to play. Confused? Let's try and explain it this way.
The Ionian mode (the major scale) of C contains the notes C, D, E, F, G, A, B. Conveniently enough, those same notes also form the Aoelian (or natural minor) scale for A, though you'd start with A, giving A, B, C, D, E, F, G. Those same notes also form the notes of the D Dorian scale: D, E, F, G, A, B, C. Get the idea? If you want to know the notes that make up a modal scale for a given note, they'll be the same as the major scale for another note.
So, for example, here are some equivalents:
C Mixolydian has the same notes as F Ionian (major)
C Lydian has the same notes as G Ionian
C Phrygian has the same notes as G#/Ab Ionian
C Dorian has the same notes as A#/Bb Ionian
C Locrian has the same notes as C#/Db Ionian
The chart gives you an easy way to remember these. See the C note? To find the scale for, say, the Mixolydian mode, you look to see where Lyd is (it's on F, one string above). That tells you that the Mixolydian scale for C has the same notes as F Ionian. So to play your groovy mixolydian improv, just choose the same patterns as you would for F major.
Learn the pattern on the chart (it's easy, only two strings to think about). Remember the names (I use the order in which they occur if you played the notes ascending): "Mix, Lyd, Phryg, Dor, Ion, Loc". Or Mix
ian. Of course, the Ionian in there isn't that useful: it tells you that to play C Ionian, you play, er, C Ionian. I left it there because it makes the pattern easier to remember.
Of course, you can move the pattern up and down the fretboard. For example, if you do it based on D (two frets up from where it's shown), it'll tell you that D Mixolydian is G Ionian, D Lydian is A Ionian, etc.
For an extra trick, if you know the natural minor scale patterns, you can use those to give you alternate ways to play the modes. For this, you need to know how to find the relative minor for any major scale. For example, the relative minor of C is A (the Am scale has all the same notes as C). If you look at the chart, you can see the "min", which gives you a way to work out the relative minor: the chart shows that for C, the relative minor is Am.
So, if you want to play in, say C Dorian, you can play (look at the chart) either Bb Ionian, or its relative minor, which would be Gm. They both contain the same notes as C Dorian.