I'll try to get a variety of styles and instruments in there because Celtic music covers everything from jigs and reels on fiddle or slow ballads sung in Gaelic or English to pipes and whistles. Where I can, I'll give a link to the artist plus imbed a Youtube sample. If you are looking for a good radio show to introduce you to a mix of the older material and the new artists, the NPR program The Thistle and the Shamrock is really good. They have a live stream of a bunch of Celtic music and a list of 100 songs they recommend here. There was a record label that produced a lot of Celtic music, Green Linnet Records. Anyone on their list of artists would probably be great.
Traditional Irish band
Battlefield Band Heave Ya Ho
Scottish traditional music
Bothy Band The Drunken Landlady
Influential traditional Irish band from the 70s.
Capercaillie Coisich A Ruin
Scottish folk band
Cherish the Ladies Inisheer
American all-female Irish-American band with leader Joanie Madden on flute
Chieftains The Frost is All Over (with the Punch Brothers)
Big kids on the block, traditional Irish band founded in the early 60s. They've done songs with everybody from Mick Jagger to James Galway.
Clannad I Will Find You from The Last of the Mohicans
Irish band whose music covers the traditional and edged into new age music.
Eileen Ivers Pachelbel's Frolics
American fiddler from the Bronx. She helped to form Cherish the Ladies.
Natalie MacMaster David's Jig/Valerie Pringle's Reel
Fiddler from Nova Scotia, Canada. She plays traditional Cape Breton fiddle music.
Solas The Newry Highwayman
Irish-American band from the US founded in 1994. If you are looking for a place to get a feel for the variety of the music, their first album Solas is my favorite and a great place to start. I love their early albums. The band lineup has changed over the years but they still sound good. A couple of the original members, particularly Karan Casey, their singer, went on to pretty successful solo careers.
Tannahill Weavers The Braes o' Balquhidder
Traditional Scottish music including bagpipes.
Just as a last bit, it's fun to see where the traditional music evolved so here are a couple of other examples.
U2 Molly Malone and Bad
There are several bands that have added punk and metal to their traditional sound. The Pogues and Floggy Molly are well-known examples and here are the Dropkick Murphys with I'm Shipping Up to Boston. It was used in the movie The Departed.
And finally, it wouldn't be a complete list without the Clancy Brothers singing the drinking song Whiskey You're the Devil.
Anyone else who happens by here and has a suggestion too, feel free to add it in the comments. Hope that gives you a place to start. Happy holidays.