I teach Celtic guitar for a living and could waffle about this subject all day, but I'll try and be brief!
As previous posters have said, Celtic music as it is played today is written in the following modes, in order of precedence:
Ionian (major scale)
Dorian (optimistic minor- compared to the major scale it has ♭3 and ♭7)
Mixolydian (blues-y major with a ♭7)
Aeolian ("Natural minor"- darker than dorian)
As harmony instruments in Celtic music are a fairly recent addition, and much of the intonation of melody instruments like violins and keyless flutes was previously based on just intonation and the overtone series, the harmony used to accompany Celtic music tends to be fairly simple. Triads and chords containing only roots and fifths are popular choices, with more complex chords generally avoided, unless any added tones within a chord are the key's root or fifth note (of course this is not always true and there are lots of jazz infused folk players, myself included).
There are seven chords available in each mode, one for each note of its scale. One of them is a diminished chord, and these are generally avoided in folk music as they sound very unstable and clash with various features of the genre.
- For ionian tunes the main chords are Ⅰ major, Ⅳ major and Ⅴ major
- For the dorian mode the main chords used are ⅰ minor, Ⅳ major and Ⅶ major or ⅴ minor
- For the mixolydian mode the main chords used are Ⅰ major, Ⅶ major and ⅴ minor
- For the aeolian mode the main chords used are ⅰ minor, Ⅳ major and Ⅶ major or ⅴ minor.
Of course many more recent players (myself included!) like to use jazzier chords and progressions based in more modern approaches to functional harmony. It could be argued that these really have very little to do with the Celtic tradition... But they are fun!
If you would like to fully understand the modes used in Celtic music and how they influence the chords one can use with different tunes, I have a free blog on the subject here: https://finaleguitar.co.uk/how_modes_work_in_celtic_music/
I have designed a wheel which shows you each of the seven triads which will fit with a melody in any given mode- they are available here: https://finaleguitar.co.uk/product/the-amazing-mode-wheel-chord-and-key-signature-finder-tool/
My book Backing Guitar Techniques For Traditional Celtic Music gives a complete guide to how you can learn to accompany this beautiful music by ear and also discusses various approaches to harmony with numerous examples, ear training exercise, chord diagrams for the guitar and lots more: https://finaleguitar.co.uk/backing-guitar-techniques-for-traditional-celtic-music/
I also have a Youtube channel devoted to the topic, Folk Friend. Many of my videos deal with how to pick out basic harmonies by ear and then use simple substitutions, bass runs and linking chords to create fluid and fitting accompaniment. Particularly you might enjoy my music theory videos which cover the complete range of chords available in each mode and are available here: https://youtube.com/video/292o7YqWGBQ/
Finally if you are interested in the physics behind the ancient "natural" scale (overtone series based) which arguably may have been widely used in ancient Celtic music then I highly recommend the work of Dr Aindrias Hirt. This probably has little to do with harmony as the likelihood is that harmony in its modern sense was not used in Celtic music at all until long after the dilution of the "natural" scale... But if you like this sort of thing, his work is fascinating and well worth a read!