Celtic music refers to the traditional or folk music that originates from countries or regions that, until a few hundred years ago, used Celtic languages. These include, especially, Irish traditional music (played throughout the island of Ireland and countries of the Irish diaspora) and Scottish traditional music (of Scotland as well as Cape Breton Island in Canada), but also Welsh traditional music (from Wales), Cornish traditional music (from Cornwall in England), Manx folk music (of the Isle of Man), Breton folk music (from Brittany, in the northwest of France), and Galician folk music (from Galicia, in the northwest of Spain). While Northumbrian traditional music is perhaps not strictly Celtic, the music of Northumbria--in the extreme north of England--shares affinities with Scottish music.
The term "Celtic music" implies that there is a single "Celtic tradition." However, virtually no one familiar with these various traditions believes that. The term is perhaps in greatest currency by music sales companies, as a catch-all genre term for music of Ireland and Scotland.
Indeed, scholars have not reached any firm conclusions about genetic relations between the traditional musics of Ireland and Scotland, on the one hand, and Brittany and Galicia, on the other. The musics of Ireland and Scotland are more closely related than any of these others, due to well-known (and ongoing) cross-pollination between the traditions. Many Irish reels originated in Scotland, and many Irish jigs and other tunes are played in Scotland. The Donegal fiddle tradition represents the coming-together of these two broad traditions.
It is extremely unlikely that modern Celtic music bears any relation to ancient Celtic music. Most of the tune types and instruments were created in recent centuries.
There is, however, a whole modern-day brand of "folk music" that uses the imagery and mythology of pagan Celtic culture, and which often goes under the name "Celtic music." This is one sort of "Celtic music" that is not specifically locatable. It may be thought of as a musical analogue to the revival of pagan religions.
Celtic music as source material for more recent branches of traditional music
Celtic fiddle tunes, along with music from England, have furnished a great deal of source material for two distinctly American branches of folk music: bluegrass and old-timey (also called old-time, or sometimes Southern or Appalachian in the context of the U.S.A.). Each of these major fiddling and singing traditions have a substantial overlap in repertoire with Celtic songs and tunes, but each branch has its own ornamentations, motifs, and performance culture that transcend stylistic differences due to locale alone and make them recognizably distinct, though related, traditions. An example of a fiddle tune that occurs in all three branches, and evolved originally in Celtic regions, is "Fisher's Hornpipe".