Answer: This is one of the highest ranked of the vexing questions in music theory classes. Everyone must come to know how to distinguish the Italian, German, and French augmented 6th chords. Not to mention knowing what an augmented sixth chord is in the first place.

It's also explained on page 67 of Exploring Theory with Practica Musica and there's an exercise in Practica Musica to help you learn them, but I'll fill you in here:

Suppose you're in the key of C. Write an Ab combined with an F# and you have two notes that are each a half-step away from G, but coming at it from different directions. Each is serving as a "leading tone" to G, and that G could be the root of the dominant chord or it could be the bass of the inverted tonic chord. Either way this interval, an augmented sixth, sets up a feeling that the next thing to happen is going to be a G.

Now you can fill out that interval in various ways, making a chord with a fuller sonority.

Add just one note, a C, and you have the "Italian" augmented 6th: Ab, C, F#.

Add C and Eb and you have the "German" augmented 6th: Ab, C, Eb, F#. This one sounds just like a dominant seventh chord built on Ab (Ab, C, Eb, Gb) except that its natural destination is G rather than Db.

If you use Ab, C, D, and F# and you have the "French" augmented 6th, whose interior major second makes it seem a little more nuanced, complex, and je-ne-sais-quoi.

Finally there's the non-national augmented sixth chord: the doubly-augmented sixth: Ab, C, D#,F#. This one moves not to G but to the inverted tonic chord G-C-E-G, which itself is a preparation for the dominant G chord. It's called "doubly-augmented" because it contains a doubly-augmented 4th (Ab-D#). Don't run into that interval very often.

Here are those chords all as they would appear in the key C:

Why the national names? No particular reason, it seems. But they had to be called something. Frank, Suzie, and Jack would have done as well.