Augmented 6th chords are voice-leading patterns more than triadic (tertial - stacks of thirds) chords. You can’t make them from stacking thirds in scales, they only exist in the voice leading of the harmonic voices. They are all chromatic pre-dominant chords targeting V, and are usually considered minor key chords. If you find them in a major key piece, it’s usually considered an example of modal borrowing (mode mixture).

There are three types:

Italian Sixth

German Sixth

French Sixth

The simplest version is the three-note Italian Sixth (It+6)

To make it, 

1. Find your V note (G if we imagine we're in C minor) and play it in octaves

2. Go up one half step from the bottom V note (to bVI or “Le”, Ab in C minor - note that this pitch is diatonic to minor harmony)

3. Go down one half-step from the top V note ("#IV" or "Fi", F# in C minor - this is the chromatic note in the chord if we are in minor to begin with)

4. Add one more note - the third above the bottom note, or you can just think of it as the "1" note - Do.

That three note chord is the It+6. If you need to double a note for four-part writing, it must be the Do, because the other two notes are active notes, sometimes called tendency tones, which shouldn't be doubled. They "want" to go to a particular resolution, so if you doubled them you'd probably wind up with parallel octaves.

To resolve the It+6, simply reverse the top and bottom notes outwards (Ab down to G, F# up to G), and then resolve the third down a half-step (C down to B natural - don’t forget to write your raised leading tone in minor!), and you are on a partial V chord. If you have two Cs, one goes to B-nat, one goes to D, now you have a complete four note V chord.

The other two Aug6 chords are four note chords. They start with this It+6 as the basis, then:

The German Sixth (Ger+6) adds a minor third above the middle note, Eb in the case of our C minor Aug6 chord. On its own it sounds like a dominant seventh chord - F# is enharmonically equivalent to Gb. But it resolves differently and is its own distinct chord (when you write it, make sure to use #IV and not bV as your top note).

There is a two step process (usually) to resolve the Ger+6:

1. Reverse the top and bottom notes back to V. Now you have a i6/4 chord.

2. Drop the middle notes a half step to so that you have a V chord.

French Sixth (Fr+6):

Start with an It+6 and add a whole step above the middle note (D if we're in C minor). All of these chords sound cool, but this one sounds especially evocative.

Resolution:

Resolves like an It+6 (reverse outer notes, drop third by a half-step). The added note is already in a V chord, so this one usually resolves straight to V.

Note that I've been referring to the "top" and "bottom" and "middle" notes, but keep in mind that like other chords, these can appear in any configuration based on the specific voice-leading desired by the composer (either to get a specific bass line or melody - see my chord progression at the top of the guide). The configuration I’ve mostly been discussing is "root position" for these chords (with le on the bottom, then do, [extra note], and fi), but again, they are more voice-leading patterns than chords in the sense we usually think of them.

Here are a couple of examples of other voicings you could use: