I don't really see what's so Elvish about Irish Music and Enya. Or maybe, yes, I see why it's associated with the Elves, but to me it's sounds too "clichéish" or "ordinary" or "soapy" to be truly Elvish. (But I actually liked the
Enya bit Guinevere linked.

) Elvish music should be something less familiar to my (western) ear, in a way something not stained by the clichéishness that now stains this Enya-style music, something that sounds more "original"...
I can't really explain, but maybe you can listen to
this. It's something I have found and associate with Elves thanks to my father, but I find it incredibly beautiful and definitely somehow Elvish (maybe more Wood Elvish than High Elvish, though.) I'm well aware that the song lasts around ten minutes, but it is really worth it to listen all of it, you can do something meanwhile and have it as background music or something, but listen to it!