What the heck are modes?
They are a system of classifying music scales that are used in traditional Celtic music.
In general, two scales are modes of each other if they have all the same notes but have different main/home notes or 'tonics'

Below I will use chord names using letters. I will signify their position on the scale with roman numerals, (II/iv), with uppercase being Major chords and lowercase being minor.

Traditional music rarely, if ever, uses the seventh (vii) diminished chord and we will ignore it.
Apologies for those not using the chrome browser. The chord charts are using CSS custom attributes. 

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. Players might call it A Dorian or A minor (though in Classical Music, the Minor, or Natural Minor, is always the 6th mode, see below)
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 key signature. If you used A key signature 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.

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

V Mixolydian

If we take the same G scale but start at the fifth note, D or V, we are now in Mixolydian 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 Mixolydian 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 key signature. If you used D key signature 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 could still use the G key signature. If you used E key signature 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.

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

 CHORDS ON THIS PAGE

MODES of the G MAJOR scale

(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 of the D MAJOR scale

(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 of the C MAJOR scale

(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 of the A MAJOR scale

(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 of the F MAJOR scale

(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