The G Augmented 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.
Gaug is G, B, D# — the 1, 3 and #5 above the root G.
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.
The shape above is movable up the neck. Every position of the chord tones on the ukulele neck — any group that sounds only G, B, D# plays the chord.
Gaug is a colour chord — it is not diatonic to a plain major or minor key, which is what gives it its character. It shows up as a borrowed chord, an altered dominant, or a passing sound.
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.