Chords:

Remember intervals? Chords are made up of more than one interval.
The most basic type of chord is called a Triad. It is made up of a tonic, 3rd and 5th. What type of 3rd and 5th depends on the scale we use to build the chord.
Lets use C Major.
CDEFGAB
T M3P5
A C Major chord uses the notes C-E-G

How about C minor?
C D Eb F G Ab Bb
T m3 P5
A Cm (remember the abbreviations?) chord uses the notes C-Eb-G

Lets put G in charge again (verb: tonicise) G A B C D EF
T M3P5
A G chord is spelled G-B-D

We name chords based on the intervals that comprise them. The name and/or abbreviation will consist of the Tonic’s pitch and the quality of the triad. The 2 main triad types or qualities are Major and minor, and are named depending on which type of 3rd the chord contains.
There are 2 derivative types of triads called diminished and augmented. Diminished chords are like minor chords, but have their 5th lowered (flatted, diminished) by a half- step. Augmented chords are like Major chords, but have their 5th raised (sharped, augmented) by a half-step.

Remember harmonic minor? CD EbF GAbB
1 M2 m3 P4 P5 m6 M7

Augmented and Diminished chords are made up from Modes of harmonic minor. Augmented chords are the result of letting 3 (Eb) be Tonic.

Eb F G Ab B C D
T M2 M3 P4 A5 M6 M7
The Eb augmented chord would be Eb-G-B. We write this as Eb aug or commonly Eb+.

Diminished chords are result of letting 7 (B) be Tonic.

BCDEFGA
T m2 m3 P4 d5 m6 d7
The B diminished chord would be B-D-F. We abbreviate this as Bdim or commonly B0

The 5th type of triad is called a suspension. Suspensions are not from the major or minor family. Remember that whether a chord is major or minor depends on the 3rds relationship to tonic. Suspensions do not contain a 3rd, rather it is replaced by either a 2nd or a 4th. Lets spell a few out on a C and G tonic.

C D EF G A B
TM2 P5
The resulting chord is spelled C-D-G and is called C suspended 2nd. We call it Csus2 for short.

GAB C DEF
T P4 P5
The resulting chord is spelled G-C-D and is called G suspended 4th. Gsus4 for short.

Notice that Csus2 and Gsus4 actually are made of the same notes? More on that later.

If you construct a chord starting on each note of a scale, you get what is called diatonic harmony.
In Cmaj
CDEFGAB

CEG= Cmaj DFA= Dm EGB= Em FAC= Fmaj GBD= Gmaj ACE= Am BDF= Bdim

It is common practice to identify harmonies by the scale degree on which they are built. A ii-V-I chord progression in the key of C would be Dm-G-C. Classical and Jazz traditions tend to use Roman numerals for this system, where uppercase denotes major and lowercase minor. Nashville and pop tend to use Arabic numbers for pitch, followed by a quality. ii-V-I = 2m-5-1. The “Nashville” system is very helpful when transposing (playing a song in a different key) as well as when improvising (soloing). You should learn this system once you have a firm understanding of scales and chords.

Extended Chords:

After we have built a triad, we can then tack on additional notes from a given scale. The most common addition is scale degree 7.
Going back to the C major:
CD E F G A B

T M3P5 M7
This would be called Cmaj7. Typically like this C^7, pronounced C Major 7

With G as Tonic:
G A B C D EF
T M3P5m7
This would be called G7. Although the m7 interval is present, we call it G7 as Gm7 would indicate a Gm triad with an added 7th. This type of 7th chord is commonly called a Dominant 7th, we’ll discuss where this name comes from later.

With D as Tonic: D EF G A B C T m3P5m7

This would be called Dm7 or D-7. Pronounce D minor 7. A dash is commonly used shorthand for minor, as is lowercase m. This chord is a minor triad with tonic D with a minor 7th above.

Am7 would mean an A minor triad with a minor 7th above. It would be spelled A-C-E-G.

We can extend even further, so lets.
First we need to extend the scale into another octave.

G A B C D EF G A T 2 3 4 5 6 7 T9

A G9 chord would start with a G major triad, G-B-D, add the m7th F, and also add the M9th A. The complete chord would be G-B-G-F-A

Numbers higher than 7 in a chord abbreviation are shorthand and always include 7.

Taken to the extreme:
G A B C D EF G A B C D E T 2 3 4 5 6 7 T 9 3 11 5 13

G13 is spelled G-B-D-F-A-C-E
Sure we could call this G7 9 11 13, but that gets a bit ridiculous, so we adhere to the following rule. Any chord extension indicated implies all lower extensions. So 13 has 11,9,7. 9 has 9 and 7.

Another common extension is the 6th chord. We build this in the same manner as a 7th, only using the interval of a major or minor 6th instead.

Inversions:

Once we construct a chord on a Tonic, we can arrange the notes in any order.
In order, a chord is said to be in root position. This is by far the most common.
With the 3rd of a chord as the lowest pitch, the chord is said to be in 1st inversion.
The 5th in the bass would be in 2nd inversion.
A very common notation for this is to name the chord in root position, then indicate the inversion with a slash followed by the note that should be positioned lowest.
A first inversion C chord would be notated as C/E. A 4th inversion C7 chord would be notated as C/Bb. Since the b7 (Bb) is spelled out directly, it is not necessary to name the chord C7/Bb, although it is also correct and accepted (I feel it is redundant).