One of the awesome things about diminished chords is that they are "mirror" chords.
The fullly diminished 7th chord is made up of three minor thirds... four notes, each a minor third away from the other. for example: gdim7 = G Bb Db Fb
Each two notes has a minor third between them, so there is also a minor third between the 7th and the root.
The awesome part is that that makes it so that even the inversions of the chord can be new chords of the same quality.
Like, in the Gdim7... G Bb Db and Fb.. if we put that in first inversion, we get Bb Db Fb and G which is now a Bbdim7 chord. we can put it in second and third inversions and get Dbdim7 and Fbdim7 chords as well.
When soloing over a diminished 7th chord, guys will often use one of two straight diminished scales... or perhaps the locrian mode, if in a major key, placing the mode in the 7th position of the major key.
The whole-half scale is pretty standard. The notes in it alternate one whole step to one half step to one whole step to one half step, and so on. If on a Gdim7 chord, we could use this scale : G A Bb C Db Eb Fb Gb G
Or, depending on your taste, the Half-Whole scale, which is the opposite: G Ab Bb Cb Db Ebb Fb Gbb
The cool thing, is that like the diminished 7th chords, these scales are also their own inversions. So, in a nut shell, You only need to learn to play 3 different scales, and you can solo over every Dim7 chord thrown at you. So, i leave you with that. goodluck.
p.s. It's been pointed out that this node is titled Diminished Triad, and i'm talking about the Diminished 7. I went into the 7th because Grimace meantioned it, and it is true that most people use the dim7 instead of just the dim triad.. almost exclusively. I hope that makes it okay.