An augmented sixth chord is a chord defined by the interval of an augmented sixth between two of it's notes. For example, an Italian augmented-sixth chord (with an A flat in the bass) consists of the notes (from the bottom up) A flat, C, F sharp. All augmented-sixth chords function as subdominant chords, and can be understood as altered inversions of sub-dominant function chords (II and IV).
Here are the four most common "sixth" chords (though there are more than these found in the literature):
(N.B. Interval designations begin from the bottom note.)
Italian = M3 (major third) + A6 (augmented sixth), derived from IV6. (scale degrees b6 + 1 + #4).
French = M3 + A4 (augmented fourth) + A6, derived from II half-diminished 4/3. (scale degrees b6 + 1 + 2 + #4)
There are two German augmented-sixth chords.
First-order German Augmented-sixth = M3 + P5 (perfect fifth) + A6 (b6 + 1 + b3 + #4), derived from IV6/5.
Second-order German Augmented-sixth = M3 + A2 (augmented second) + A6 (b6 + 1 + #2 + #4), derived from II half-diminished 4/3.
The second-order German augmented-sixth chord, which utlizes the interval of an augmented second, is constructed to lead specifically to the major tonic through the normative dominant suspension (V6/4 - V 5/3).
The etymology of all these names is unclear. All augmented-sixth chords derived from subdominant-function Stufen of the tonic key have a flattened submediant, that is the sixth degree of the scale (e.g., A flat in C major), and a raised fourth (F# in C major). These specific scale degrees, however, will not obtain in all cases (e.g., applied-functions within subordinate prolongations found in lower Schichten, such as #IV7/IIb).
Augmented sixth chords usually have the flattened sixth as the bass note. When they do, they resolve to the dominant suspension (V6/4) in order to avoid parallel fifths and stand as an example of chromaticism.