If we then raise the tonic note (by an augmented unison), the interval between the bass note and root note becomes an augmented sixth. A chord with this interval of an augmented sixth is called an Augmented Sixth Chord.
The three basic types of Augmented sixth chords:
An Italian Sixth Chord has an augmented sixth between the bass and root of the chord, with the fifth of the chord in-between the bass note and root.
A German Sixth is like the Italian sixth but with one extra note placed a perfect fifth above the bass note.
A French Sixth is like the Italian sixth but with one extra note placed a Augmented fourth above the bass note.
The resolutions of Augmented sixth chords:
Italian and French Sixth chords will most often resolve to a dominant chord.
The German Sixth will most often resolve to a dominant or tonic chord. (It is worth noting that if the German Sixth resolves to the dominant then parallel fifths will occur, which can cause musical lines to lose their independence in certain styles of music.)