An Irishman is a man from Ireland — or, depending on the speaker, a man whose family is Irish. It is a plain joining of Irish and man. You will see it in stories, older writing, and news. Today some people prefer “an Irish man” as two words, or “an Irish person,” because Irishman can sound a little like a ready-made type, as if one word could stand for a whole people. It only names a man; it does not cover Irish women. Keep the capital I, because Irish is a proper adjective. Though actually, many Irish people would rather you just say he is Irish.