In Gaelic usage, "Black Irish" has nothing to do with being mixed-race or one's religion.
In Ireland, it traditionally means an Irish person who has dark hair and eyes, though some also apply it to those with lighter eyes. This usage has continued in the diaspora among those who are still culturally Irish and Scottish.
The WP article is not too bad:
http://en.wikipedia.org/wiki/Black_Irish, and touches on how language does evolve over time and these things can become more complex... but a small number of people misusing a term doesn't change the historical/majority usage.
Irish and Scottish Gaelic nicknames often use hair color as a prominent descriptor, and while the literal translation of a nickname like "Seán Dubh" is "Black John", and someone with no Gaelic might think it means Seán/John is what we think of as a Person of Color, it really only means the guy has dark hair (and carries the assumption he's a white (Irish) guy with dark hair).
A film that touches on this is
The Secret of Roan Inish, where the family lore is that "the dark ones" in the family are descended from the Selkies (the shape-changing seal people).
As a Chatholic, being raised in the Chatholic faith and still a practicing Chatholic I can safely say that the term Black Irish means a Protestant Irish person. And no I'm not Irish, but have been told as a child by Nuns and other Chatholics alike what the term means.
Hi Diana, may I ask where and when you heard this? I am an American of predominantly Irish and Scottish descent, the usual product of the famines and the clearances. I grew up in a mixed Catholic/Protestant diasporic community in Northern Illinois, and as a young adult lived in Irish immigrant communities in Boston and Chicago. While some Irish people would use "Black" as a slur, and I guess some Catholics might say it about Protestants (and vice versa) because of this, I don't recall anyone ever saying "Black Irish" simply meant Protestants.
While among the Irish "Black Irish" does not mean mixed-race, it is a misconception I've heard multiple times in America. The English in particular compared the Irish to Africans, and Cromwell enslaved them along with the Africans in the Caribbean, so there were racist caricatures drawn of Irish people where they have more African features. This was done by people who were oppressing the Irish as well as People of Color. Google "Anti-Irish racism" for some of that. Slán.