Add a 6th or some kind of 7th to the triad and you have a four note chord.
C6
6 says add the diatonic 6th to the chord.
C6 = C triad and diatonic 6.
| diatonic scale C | = | C | D | E | F | G | A | B | C |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||
| chord C | = | C | E | G | |||||
| chord C6 | = | C | E | G | A | ||||
| chord Cm | = | C | Eb | G | |||||
| chord Cm6 | = | C | Eb | G | A |
C7
A naked 7 in the chord symbol is a dominant 7th. Add the dominant 7th to the triad.
V7 = dominant 7.
C7 = C & V7
Cm7 = Cm & V7
Cm7b5 = Cmb5 & V7
C7+ = C+ & V7
C6/7 = C & 6 & V7
| diatonic scale C | = | C | D | E | F | G | A | B | C |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||
| chord C7 | = | C | E | G | Bb | ||||
| chord Cm7 | = | C | Eb | G | Bb | ||||
| chord Cm7b5 | = | C | Eb | Gb | Bb | ||||
| chord C7+ | = | C | E | G# | Bb | ||||
| chord C6/7 | = | C | E | G | A | Bb |
Cmaj7
Cmaj7 = C & maj7. Add the major 7th to the triad.
| diatonic scale C | = | C | D | E | F | G | A | B | C |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||
| chord C | = | C | E | G | |||||
| chord Cmaj7 | = | C | E | G | B |
There are many ways to write the major 7th.
For example 
, 
and C(#7) all mean Cmaj7. The natural 7th indicates the major 7th is true to the diatonic scale. The #7 indicates the major 7th is a semitone higher than the dominant 7th.
We prefer the maj7 notation as we feel it is the most self explanatory.
Cº is C dim is C diminished.
The formula for a diminished chord is: minor 3rd, flat 5th, diminished 7th.
| diatonic scale C | = | C | D | E | F | G | A | B | C |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||
| chord C | = | C | E | G | |||||
| chord Cm | = | C | Eb | G | |||||
| chord Cmb5 | = | C | Eb | Gb | |||||
| chord Cdim | = | C | Eb | Gb | Bbb |
Bbb is enharmonic to A (the sixth of C). Cdim is the correct way to write Cm6b5.
Cmb5 is sometimes called a half diminished chord. It is on the road to diminished. C half diminished is written as 
Four note chords in summary
| diatonic scale C | C | C# Db | D | D# Eb | E | F | F# Gb | G | G# Ab | A | A# Bb | B | C |
| Root | S | T | m3 | maj3 | 4 | b5 | 5 | + | 6 | V7 | maj7 | 8ve | |
| chord C6 | C | E | G | A | |||||||||
| chord Cm6 | C | Eb | G | A | |||||||||
| chord C7 | C | E | G | Bb | |||||||||
| chord Cm7 | C | Eb | G | Bb | |||||||||
| chord Cm7b5 | C | Eb | Gb | Bb | |||||||||
| chord C7+ | C | E | G# | Bb | |||||||||
| chord C6/7 | C | E | G | A | Bb | ||||||||
| chord Cmaj7 | C | E | G | B | |||||||||
| chord Cdim | C | Eb | Gb | BBB |