Anyway, the half diminished scale (though it isn't usually called that) would be the scale associated with the half diminished chord, which is a minor7b5 chord. This is the locrian scale. It is spelt R b3 b5 b7. While it has the two minor third intervals, it also has another major third interval on top of those, making it not fully diminished. The locrian scale is spelled R b2 b3 4 b5 b6 b7.
There are two diminished scales, and while you could think of them as being built on the diminished triad, with those two minor thirds, i think that it's more useful to think of them as coming from the fully diminished 7 chord, spelled R b3 b5 bb7. They are the half-whole and the whole-half scales. Their names explain how they are built, the half whole scale consists of repeating the pattern half step whole step half step, etc. In C it would be C, Db, Eb, Fb, etc. The whole half step is similar, but with a whole step interval first. In C: C, D, Eb, F, etc. They are also called octatonic scales, because there are 8 notes to the octave instead of 7 as there are in regular diatonic scales.
As for use, locrian is used over m7b5 chords (often in the context of a minor ii V I.) The octatonic scales are used over diminished chords, and the half whole is used over many altered dominant chords. Ever see the hendrix chord (E7#9) and not know how to solo over it? Try using the E half whole diminished scale; it will fit and sound interesting.
