The C Half-Diminished chord and everything around it — the notes and how they are built, the shape on piano and guitar, the sound, and the keys it belongs to.
Cm7♭5 is C, Eb, Gb, Bb — the 1, b3, b5 and b7 above the root C.
Root notes are red. The shapes above are movable — an E-shape has its root on the low E string, an A-shape on the A string. any group of these notes on the neck sounds the chord.
Every position of the chord tones on the ukulele neck — any group that sounds only C, Eb, Gb, Bb plays the chord.
Work out the chords of a whole key with the Scale & Chord Finder, name a chord from its notes with the Chord Identifier, or browse every chord tool. The theory behind how chords are built is in the free Cloud Atelier music guides.