"Celtic" doesn't refer to any one group, language, ethnicity, etc., in particular (except maybe one tribe in Gaul that Caesar had dealings with); rather it's a catch-all term for... well, that's the question. People who speak Celtic languages, obviously. Who are they? The Celts. How do we know? Because they speak Celtic languages. I'm not a historian though - that was just my editorialized take on what a confusing word it is.

I did find this AH thread that compiles several different answers, some directly related to the question, others more tangential.