In studying music theory I have heard the term Augmented 6th quite a few times, but I'm not quite sure what it is. What is a Augmented 6th and how is it used in a composition?
Augmented sixth chords? The bass leads by semitone down to the root of the next chord; the ascending leading tone to the root of the next chord is also present in one of the upper voices. The notes are really a minor seventh apart, but, because they are written as the second and seventh degrees respectively relative to the root of the next chord, the interval formed is written as an augmented sixth.
As an example, the most basic augmented sixth chord is an Italian sixth. Used as a predominant in the key of C, that is, leading to a G major chord, it would be A♭ in the bass with C and F♯ in the upper voices. The interval A♭ to F♯ is the augmented sixth. The other commonly used forms of augmented sixth chord are the French sixth, which in this example would add D to the Italian sixth's upper voices, and a German sixth, which would add E♭ instead.
The Wikipedia article is fairly comprehensive.