What is in it
A half-diminished chord is a diminished triad (root, minor 3rd, diminished 5th) with a minor 7th added above. From C that is C, E flat, G flat, B flat. The diminished triad at the bottom supplies the flat 5th and its tension; the minor 7th on top is what makes the chord half-diminished rather than fully diminished.
On the staff
Tense, but not chaotic
The bare diminished triad is sharply unstable. Adding a minor 7th to it does not remove the flat-5th tension, but it cushions it. The 7th gives the ear another, more consonant interval to hold onto, so the chord reads as coloured and unsettled rather than urgent. It is tension as atmosphere. The lead-sheet name, m7 flat 5, captures it well: a minor 7 chord with the 5th pulled flat, dark and slightly off balance but still musical to sit on.
Where you hear it
In a major key the half-diminished is the seventh-chord form of the leading-tone chord, vii ø7. Its real home, though, is the minor-key ii-V-i, where it is the ii chord, the standard way jazz sets up a minor resolution. Reach back far enough and the same sonority opens Wagner's Tristan und Isolde, the "Tristan chord" that helped pull Romantic harmony loose from its moorings.
Jazz treats it as a scale as much as a chord. Mark Levine notes that bebop players read it through the Locrian mode, while from the 1960s on the preferred choice became the sixth mode of the melodic minor scale, which keeps the flat 5th but raises the 2nd to a brighter note. In a minor ii-V-i the half-diminished ii is followed by an altered dominant and a minor-major tonic, three chords Levine draws from three different melodic minor scales.
How Aubel handles it
Half-diminished unlocks eighth. It takes the diminished triad you trained as a plain chord earlier and adds a minor 7th, so the cue is the flat 5th you already know, now under a 7th that softens rather than sharpens the tension. The contrast to listen for is against the fully diminished 7, which lowers that top note further. Diminished 7 unlocks last.