As an American of Irish ancestry, I view the word "Irish" as meaning a person born in Ireland, or an Irish national. Just as I would with "Japanese" or "Welsh".

The very recent tendency of Americans to hybridize their identity, by using their ancestry hyphenated with the word "American" I've always thought an odd way of distinguishing one's self as "not mainstream WASP".

That said, growing up, when someone asked me "what I was" I never had any problem figuring out what they were asking--it was either my ancestry or my religion.

Just for the record--it wasn't nearly as trendy to be of Irish ancestry and a Catholic in the US prior to the 90s Irish fads.