The modes
If the third is minor (1–m3–P5), we say the triad is minor and write it with a lowercase roman numeral, where the number is the position of the root note (1) in the sequence of notes in the scale. If the third is major (1–M3–P5) then it’s called major, and we write it with an uppercase roman numeral. And if the fifth is diminished (1–m3–d5), we call it minor diminished (it cannot be a major diminished chord, because there is only one tone between the major third and the diminished fifth!). For example, if we are playing with the scale of white keys with the root note being C, and we play a triad on the third note E, we play E–G–B, which is E minor written 'iii'.
The Ionian mode (or "major scale")
What's special about the major scale?
For those intervals that can be major or minor (second, third, sixth, and sevenths), are all major! It really makes sense to call this scale "Major" (I've not heard it described this way). The proper name of that scale is Ionian, and we can call it "the most used scale in Western music", for good reasons that blend acoustics and mathematics, and creative convenience.
To make it even more "major" sounding, its fourth and fifth notes' triads are major chords. Since the fourth and the fifth notes are commonly used as roots for chord progressions, there's something harmonious about that scale that explains why it is so popular.
And the minor scale is also special in the same way?
Almost. There is a scale where all the intervals are minor: it's Phrygian! But... why don't we use it much in Western music? Why is the minor scale not the Phrygian scale, but the Eolian scale? I'm not a scholar, but I suspect it's because the chord on the fifth (vº) is diminished in Phrygian. If you want to play other chords, you would have to avoid the dominant (fifth note), which is unusual; a common way to finish a song is to go from the dominant to the tonic (first note), so... Phrygian is not used much. We eliminate it from our list of Celtic sounding scales.
What can we do to have a perfect fifth, while keeping as many minor intervals as possible? We switch to Aeolian! It has a major second, which is what was needed to make the fifth a minor chord – and now its fourth and fifth notes are roots of a minor chord. This is probably why Aoelian is the basis for the minor key.
What do you mean "basis"?
The minor scale has four variants. Technically, Aeolian is "natural minor" and "melodic minor descending". "harmonic minor" is TSTTS+T (Asymmetry! Imbalance! Oh no!). "Melodic minor ascending" is TSTTTTS (More asymmetry! More imbalance! Boooo!).
This to remind us that all our fancy rules are meant to be thrown away at the first opportunity because "oh yeah, it sounds cool when you do this".
Mixolydian!
Dorian!
Other scales!
Why does Celtic music use all of Ionian, Myxolydian, Dorian, and Aeolian, but not Lydian, Phrygian, Locrian?
We eliminated Phrygian earlier as being not good enough to be our minor scale, because of the diminished (tritonic) triad on its fifth note, which restricts the types of chord progressions you can use.
There are no perfect fourths in Lydian and no perfect fifths in Locrian, so they don't stand out as being particularly harmonious. It's no surprise they are not used in Celtic music.
Myxolidian and Dorian are steps in between Ionian (the majorest scale) and Aeolian (the minorest scale). You can see Myxolydian as a way to blend some characteristic of minor into the major scale, without introducing the minor third, which is the telltale sign of a minor key when you play a triad on the root note. This is done by using the minor seventh, which has this sadder, more heartfelt feeling compared to the triumphant, joyful major seventh, especially when played when resolving a phrase to the tonic (I).
You can see Dorian as a way to blend some characteristic of major into the minor scale, introducing a major 6th. There's ways to arrange minor scale music to be optimistic (instead of gloomy) and the major 6th contributes to that feeling.