Neapolitan sixth chord is a major triad formed on the flatted second degree of the scale, so it would be, in the key of c minor, a Db major chord. It is called a Neapolitan sixth because it is usually used in first inversion (ie with the third of the chord in the bass), which would have been indicated in figured bass as a 6 under the bass line. You can find an example in measure 3 of Beethoven's Bagatelle opus 119, no. 9, where you'll find a first inversion Bb major chord in a piece opening in a minor.
The Italian augmented sixth consists of the tonic, the augmented fourth and the flatted sixth. There's an example in measure 12 of the Overture to the Magic Flute.
The French augmented sixth chord is the Italian augmented sixth plus the second degree of the scale. You can find an example in measure 38 of the minuet from Beethoven's Piano Sonata Opus 10, no. 3.
The German augmented sixth chord is the Italian augmented sixth plus a minor mode third degree of the scale. You can find an example in measure 33 of the second movement of Beethovens first String Quarter, Op.18, no. 1.
All this I found in Kostka and Payne, Tonal Harmony.