The word augmented refers to a type of interval and augmented chords are chords that contain that augmented interval above the chord root.
An augmented interval is a perfect or major interval that has been stretched or expanded (augmented) by a semitone (but keeping the same letter names).
C - G is a perfect fifth: C - G# is an augmented fifth
C - F is a perfect fourth: C - F# is an augmented fourth
C - D is a major second: C - D# is an augmented second
C - A is a major sixth: C - A# is an augmented sixth.
The one most commonly found in chords is the augmented fifth but the 7th aug 9 chord contains an augmented 2nd. Chords named as something #11 contain an augmented 4th above the root although the chord is called sharp 11 rather than than aug 4 or aug11
The augmented sixth chord doesn't appear in chord notation as it's a feature of individual moving parts (think barbershop harmony).
In its most common form, it sounds exactly the same as the dominant 7th:
C aug 6 = C E G A#
C dom 7 = C E G Bb
Out of context, they are exactly the same. The difference lies in how their individual notes connect to the notes of the next chord. (A# wants to rise a semitone, Bb wants to fall a semitone).
More precisely:
The aug 6 interval, C - A#, wants to expand to an octave: B - B (you can find it in songs in the key of E, written as C7 - B major)
The minor 7th interval, C - Bb, wants to shrink to a major third: F - A (It's part of the common V7 - I, or C7 - F in the key of F)
As chord notation doesn't specify how the individual notes of chords are supposed to connect to each other, there's no need for it, so it just appears as a 7th chord. It does appear in standard notation scores though.
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