Augmented sixth chords usually occur in minor keys, though they can occur in major keys, and are built upon the lowered 6th scale degree. They all include the lowered 6th scale degree which resolves down to the dominant by a half step, and a raised 4th scale degree which resolves up to the dominant by a half step. They also include scale degree 1.

The Italian Sixth Chord includes three notes: b6, 1, and #4, and these notes resolve to a dominant without a fifth. While functional, this resolution sounds more empty in my opinion.

The German Sixth Chord contains all of the Italian Sixth notes, but also contains a b3. This lends itself very naturally to a i 6/4 – V resolution (to avoid parellel fifths), which is my favorite sixth resolution.

The French Sixth Chord contains all of the Italian Sixth notes, but also contains the supertonic. This presents a chord built on two tritones, which gives it a less settled, mysterious quality.

Additional information from tobyrush.com: