In the history of Western music the dominant seventh was the first discordant chord to be used as freely as the consonant major and minor triads. It is also the first discordant chord that most people come across in their theory instruction.
For this reason it is a very well understood chord. The augmented sixth chord, however, is very poorly understood despite its clear similarity to the dominant seventh chord. It can be most easily understood by examining it in parallel to the dominant seventh type chord, which is how I shall approach it here.
The variety of names under which the augmented sixth chord has laboured also helps to obscure its understanding - in classical theory it is often referred to as the French sixth, the German sixth, or the Italian sixth. In jazz this chord is most frequently described and understood as a tritone substitution (of a dominant chord). The three classical names all refer to chords containing a root, a major third, and an augmented sixth, but differ in their choice of additional pitches: the German contains a perfect fifth, the French contains an augmented fourth, the Italian contains has no fifth or fourth (in four-part harmony, the third is usually doubled). Despite these differences the chords are functionally identical.