In studying music theory I have heard the term Neapolitan 6th quite a few times, but I'm not quite sure what it is. What is a Neapolitan 6th and how is it used in a composition?
It is a Major triad built on the lowered 2nd scale degree. It's usually in first inversion, hence the "6th" part of the name. So if I'm in C-minor, the Neapolitan 6th (sometimes analyzed as N6 or bII6) would be a Db-major triad, probably with the F in the bass. They are chromatic harmonies, and their primary function is to go to V.
EDITED TO ADD: There is a very particular voice-leading involved also, usually the voice that has the b2 note moves directly to the leading tone in the upcoming dominant harmony. This is a rare example of a diminished melodic third being standard practice.
A good example is in beats 3 and 4 of the 3rd measure of the first movement of Beethoven's "Moonlight" Sonata. The piece is in c#-minor, and the 3rd measure starts on A (VI) and moves to a D Major chord in first inversion (the Neapolitan). The D-natural root of that chord is chromatic to the key, and is what causes the harmony's somewhat exotic quality. It's an alteration of the diatonic d#-dim (iio) chord that could just have easily have been used here, but would have been far less dramatic. Either way, the next harmony is V (with some cadential 6/4 action thrown in for seasoning).