Below are a few third chord progressions (or part of progressions).

Roman numeral 'I' represents the major chord of the root note.

In the scale of C, C major chord is the root chord – written as I (first) chord in progressions.

'ii' represents the second note in the scale. the small letter indicates minor quality of that position chord. If it was 'II' it would mean a major chord.

iim | V| Imaj

In the scale of C, C major chord is the root chord –  I (first) chord

Write the Triads from each note of the C major scale : C D E F G A B C

(start from each note, skip the next note in the scale, take the next note, skip the next note, take the next – total three notes – forms a triad)

1. CEG (C major chord)

2. DFA (minor chord)

… etc

5. G B D (G major chord)

We find that the chord starting from the second note of a major scale is a minor chord, the fifth is a major and from the root is major chord.

For the scale of C major the progression iim | V| Imaj  will be Dm | G | C.

For D as root. Same way if I write a turn around for D : fifth of D is A, second is E minor. Fifth is A major.

The progression 2 5 1 turn around for root D will be Em |A | D

Why do we have to learn about these three chords?

We find that this chord progression in its most basic form as above or in its variations (discussed below) are found in a wide majority of jazz, pop, western classical songs.

Theory comes after music. We try to find out why something works in music and note it down as theory. This particular three chords have been used a lot because it worked to give some very appealing effects.

The iim is an addition to the 'perfect cadence' that V to I motion gives. (e.g. of V to I is Gmajor to Cmajor chord). V to I is a very obvious motion used in chords to come back to root – resolution after the tension given my the 5th chord (G in the case of C)

For a complete step by step guide to understanding and using chords, see The Chord Code

iim | V7 | Imaj

Changing the fifth position major chord of the 2 5 1 to the seventh chord we get iim | V7 | Imaj

Find the seventh chord for the fifth position chord.

In the scale of C, G is the fifth note, G B D is the triad, now add the 7th note with respect to G, counting within the scale of C.

C D E F G (1) A(2) B(3) C(4) D(5) E(6) F(7) G

The seventh note from G is F.

GBDF will be the notes in a seventh chord of G as the fifth position chord.

F is a minor 7th note when G is the root. (Music theory for beginner musicians). Major 7th note with respect to G would be F# which is not in C major scale – c major scale has no flats or sharps.

GBDF in terms of intervals : G – root     B – major 3rd     D – fifth     F – minor 7th

Major 3rd and minor 7th intervals in a chord gives a dominant seventh chord (the chord usually called 'the seventh' chord).

So GBDF is the G dominant seventh chord or the G seventh chord, written as G7

iim | V7 | Imaj for C is Dm G7 C

Iim7 | V7 | Imaj7

Find 7ths for the second chord of C major scale and the first chord, the root chord.
Take the root chord. CEG add the seventh note B. B is a major 7th note when C is the root. major 3rd + major 7th = Major 7th chord
The second chord. DFA, seventh note is C. C is minor 7th note when D is the root. Minor 3rd note + minor 7th note = minor 7th chord

iim9 | V#9♭13 | Imaj9

We can add other possible notes to the chords and embellish them.

Iim9th – minor 7th chord + major 2nd note (9th note)         Dm9th : Dminor7th + E (major 2nd)

I maj 9 – major 3rd major 7th + major 2nd note                   Cmaj9th : Cmajor 7th + D

V#9♭13 – sharp ninth flat 13th chord

#9 note is same as #2 – the note after the major 2nd note. For G #9 or #2 is the note after A (major 2nd) – which is A#. (G-root, G# – minor 2nd, A – major 2nd, A# – #2 or #9)

♭13 is same as ♭6 (any number above 7, just subtract 7 from it and it is the same note).

5th for G is D, 6th is E (D# will be #5th note) ♭6 will be one note before 6th – one note before E which is E♭ (same as #5 – D#. D# and E♭ are enharmonic – the same note)

So G#9♭13 has the following notes GBDFA#E♭

Check out The chord code to make life easier with chord symbols and names and to learn what chords are made of.