Augmented sixth chord
The augmented sixth chord is a chord that is primarily used as a strong preparation for a dominant chord. The augmented sixth chord comes in several varieties, but all of them share three common tones. The augmented sixth chord is characterized by the augmented sixth interval that is formed between the notes one half-step above and one half-step below the dominant tone of your key. Apart from those two tones, the third tone found in all augmented sixth chords is the tonic note. For example, a basic augmented sixth chord in the key of C major or minor would consist of an A♭, a C, and an F♯. An augmented sixth chord gets its emphasis from the two "outside" notes (In the previous example, A♭ and F♯) both chromatically approaching the dominant tone, G. In addition to these three notes, there are three variants of the augmented sixth in which a fourth note is added.
Standard harmonic function
From the Baroque to the Romantic period, augmented sixth chords have had the same harmonic function: as a chromatically altered predominant chord leading to a dominant chord. During the Romantic period, the use of the augmented sixth chord increased in ambiguity as composers explored other functional possibilities outside of its role as a predominant. Since a chord could simultaneously have more than one enharmonic spelling with different functions (i.e., both predominant as a German sixth and dominant as a dominant seventh), its function could be reinterpreted mid-phrase. This heightens both chromaticism by making possible the tonicization of remote keys, and possible dissonances with the juxtaposition of remotely related keys.
Augmented sixth chords are occasionally used with a different chord member in the bass. Since there is no consensus among theorists that they are in root position in their normal form, the word "inversion" isn't necessarily accurate, but is found in some textbooks, nonetheless. Sometimes, "inverted" augmented sixth chords occur as a product of voice leading.
Italian augmented sixth
The Italian sixth has only the three basic tones, in the key of C: A♭, C, and F♯. The note C is doubled, being played by both the alto and tenor voices which resolve in opposite directions.
German augmented sixth
The German sixth has different four tones. In addition to the previous three tones listed, the German sixth contains the pitch a minor 3rd above your tonic pitch. For example, in the key of C major or minor, a German sixth would contain an A♭, a C, an E♭, and an F♯.
French augmented sixth
The French sixth also is a four-tone chord. Besides the three tones of the Italian sixth, the French sixth contains the pitch a major 2nd above your tonic pitch. For example, in the key of C, a French sixth would contain an A♭, a C, a D, and an F♯.