This is the ordinary country adjective, and it stretches over everything that country touches: a passport, a family name, a song, a kind of bread, a dance, a football team, a way of speaking. It always begins with a capital I, wherever it sits in the sentence, because it is built on the name of a place. It goes in front of a noun (an Irish song) and after be (her mother is Irish). It does not change for one person or many. And note that it covers the language too, so Irish music and an Irish word are both normal.