What the heck are modes?
They are a system of classifying music scales that are used in traditional Celtic music.

Below I will use chord names using letters but also their position on the scale with roman numerals, where Capital letters signify Major chords and lowercase signify minor chords.

Traditional music rarely, if ever, uses the 7 diminished chord and we will ignore it.

I Ionian

Let us take a G Major scale, with its intervals of tone | tone | semi-tone | tone | tone | tone | semi-tone
the notes are: G — A — B C — D — E — F# G 
The chords we could play for a tune in G Major include
G Major — A minor — B minor C Major — D Major — E minor

Using roman numerals:
I ii iii IV V vi 

The melody may use a lot of G notes and often will end on an G note.
We could call this the First Mode of the G scale. We also call it Ionian. Major Scale = Ionian Mode.
In Ionian, the I, IV, and V chords are very common, you will also hear the ii and vi . The iii minor (sometimes major) is less common but cool when you hear it.

II Dorian

If we take the same G scale, the same notes and the same intervals but start at the second note, A or II, we are now in Dorian Mode
and the notes are A — B C — D — E — F# G — A.
We could call this the Second Mode of the G scale. We can also call it A minor or A Dorian.
The melody may use a lot of A notes and often will end on an A note.
If you want to write the sheet music, you would use the G scale. If you used A Major you would have to use a flat-3 and a flat-7.
In Dorian mode (start counting from the A)  the i, VII chords are very common. You will also hear the v and sometimes the IV.

Trad music rarely uses the Mode that starts on the III, called Phyrgian.
Trad music rarely uses the Mode that starts on the IV, called Lydian.

V Myxilodian

If we take the same G scale but start at the fifth note, D or V, we are now in Myxilodian Mode
and the notes are D — E — F# G —A — B C — D.
We could call this the Fifth Mode of the G scale. We can also call it D Myxilodian or D Mix.
The melody may use a lot of D notes and often will end on an D note.
If you want to write the sheet music, you would still use the G scale. If you used D Major you would have to use a flat-7.
In Myxilodian the I and VII chords are very common. And you will hear the v.

VI Aeolian

If we take the same G scale but start at the sixth note, E or VI, we are now in Aeolian Mode
and the notes are E — F# G —A — B C — D — E.
We could call this the Sixth Mode of the G scale. We can also call it E minor or E Aeolian.
The melody may use a lot of E notes and often will end on an E note.
If you want to write the sheet music, you would still use the G scale. If you used E Major you would have to use a flat-3, flat-6 and flat-7.
In Aeolian the i and VII chords are very common but also any of III, iv, v, VI.

Trad music rarely uses the Mode that starts on the VII, called Locrian.

 CHORDS ON THIS PAGE

MODES based on G MAJOR

(Modes not common in Irish Trad: Phyrgian III, Lydian IV, Locrian VII)

IONIAN I on G (G Major)

G — A — B C — D — E — F# G

DORIAN II on G (A Minor) (A Dorian)

G — A — B C — D — E — F# G — A

MYXILODIAN V on G (D Mix)

G —A — B C — D — E — F# G —A — B C — D

AEOLIAN VI on G (E Minor) (E Aeolian)

G —A — B C — D — E — F# G —A — B C — D — E

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MODES based on D MAJOR

(Modes not common in Irish Trad: Phyrgian III, Lydian IV, Locrian VII)

IONIAN I on D (D Major)

D — E — F# G — A — B — C# D

DORIAN II on D (E Minor) (E Dorian)

D — E — F# G — A — B — C# D — E

MYXILODIAN V on D (A Mix)

D — E — F# G — A — B — C# D — E — F# G — A

AEOLIAN VI on D (B Minor) (B Aeolian)

D — E — F# G — A — B — C# D — E — F# G — A — B

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MODES based on C MAJOR

(Modes not common in Irish Trad: Phyrgian III, Lydian IV, Locrian VII)

IONIAN I on C (C Major)

C — D — E F — G — A — B C

DORIAN II on C (D Minor) (D Dorian)

C — D — E F — G — A — B C — D

AEOLIAN VI on C (A Minor) (A Aeolian)

C — D — E F — G — A — B C — D — E F — G — A

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MODES based on A MAJOR

(Modes not common in Irish Trad: Phyrgian III, Lydian IV, Locrian VII)

IONIAN I on A (A Major)

A — B — C# D — E — F# — G# A

DORIAN II on A (B Minor) (B Dorian)

A — B — C# D — E — F# — G# A — B

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MODES based on F MAJOR

(Modes not common in Irish Trad: Phyrgian III, Lydian IV, Locrian VII)

IONIAN I on F (F Major)

F — G — A Bb — C — D — E F

DORIAN II on F (G Minor) (G Dorian)

F — G — A Bb — C — D — E F — G

AEOLIAN VI on F (D Minor) (D Aeolian)

F — G — A Bb — C — D — E F — G — A Bb — C — D