4. Seventh Chords

Seventh chords are triads with an added fourth note. There are a few types. (The more important ones are starred)

(Extended aug chord)
Augmented major seventh (1-M3-#5-M7) – denoted “C+M7”, "C∆+", "CaugM7"
Augmented seventh (1-M3-#5-m7) - denoted "C+7", "Caug7"

(Extended maj chord)
*Major seventh (1-M3-5-M7) – denoted “C Maj7”, "CM7", "C∆7", "C∆"
*Dominant seventh (1-M3-5-m7) – denoted “C7”

(Extended min chord)
Minor/major seventh (1-m3-5-M7) – denoted “Cm/Maj7”
*Minor seventh (1-m3-5-m7) – denoted “Cm7”

(Extended dim chord)
Half-diminished seventh (1-m3-b5-m7) – denoted “C∅”
Full-diminished seventh (1-m3-b5-dim7*) – denoted “C°7”, "Cdim7"

(*a diminished 7th is the same as a Major 6th. More on this later.)

Also, note the intervals between the notes of the chords:
m7 "1-(m3)-> m3 -(M3)-> 5 -(m3)-> m7"
m/Maj7 "1-(m3)-> m3 -(M3)-> 5 -(M3)-> M7"

7 "1-(M3)-> M3 -(m3)-> 5 -(m3)-> m7"
Maj7 "1-(M3)-> M3 -(m3)-> 5 -(M3)-> M7"

∅ "1-(m3)-> m3 -(m3)-> b5 -(M3)-> m7"
°7 "1-(m3)-> m3 -(m3)-> b5 -(m3)-> dim7*(M6)"

DIATONIC 7th CHORDS:
There is also a diatonic seventh chord for each note in a given scale. In C major they are C Maj7 (CEGB), Dm7 (DFAC), Em7 (EGBD), F Maj7 (FACE), G7 (GBDF), Am7 (ACEG), and B∅ (BDFA). Major scale seventh chords always go (M7/m7/m7/M7/7/m7/∅).

Seventh chords also have inversions. Root (1-3-5-7), 1st (3-5-7-1), 2nd (5-7-1-3) and 3rd (7-1-3-5).

FULL-DIMINISHED CHORDS:
Notice anything funny about full-diminished chords? Invert one, what do you get? A different full-diminished chord! There are really only three different °7 chords. Let's look at C°7 (C-Eb-Gb-Bbb).

["Bbb" is the same as "A", but since it's our seventh in this instance it should have a "B" name. Don't get hung up on it.]

So really we have (C-Eb-Gb-A). Let's invert.

1st inversion (Eb-Gb-A-C)
2nd inversion (Gb-A-C-Eb)
3rd inversion (A-C-Eb-Gb)

All three are "°7" chords, because all their notes are still minor thirds apart from each other. °7 chords have this cool overlapping, symmetry kinda thing going for them. (Possible segue for key transitions?)