What is a Half Diminished Chord?
It is basically a blend between a minor 7th chord and a diminished 7 chord.
You start with a standard minor 7th chord, then you flatten the 5th, so the chord can be labeled: m7(b5).
For example, start by playing a C minor 7th chord, take the 5th and lower it a half-step, which makes it a C min7(♭5), which can also be labeled in short as: Cø7. The chord formula is therefore: 1, ♭3, ♭5, ♭7.
C Half Diminished 7th Chord: C min7(♭5)
How to use Half Diminished Chords?
This chord can be used in so many ways since it has several different chord qualities. The 3rd is minor so it has an emotional quality, the 7th is also minor which adds another dark quality, and the 5th is a tritone which adds tension.
Half diminished chords can be excellent as transition and passing chords for adding movement to the voice leading. They can also be used to add tension while still not being fully diminished.
I personally like to throw them in sometimes for adding some non-diatonic tension in the chord progressions. Because overall it is a dark and emotional chord, with a mysterious and strange vibe, yet a distinct minor quality.