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Irish, of course. A famous Irish family originating in Co. Leitrim, where they were the ruling clan. The name is one of the few with the O prefix which are not Celtic. The given name Ruarc, whence O Ruairc, is from the Old Norse Hrothekr. In England and Scotland that became Roderick, in Russia Rurik.
That's not to say that the O'Rourkes are of Norse origin, their Irish pedigree is recorded from far back in time. I'm sure if you do a websearch on O'Rourke you'll learn a lot more.