On ZENIT there is an article on Archbp. Martin of Dublin, the sexual abuse scandal, and Church and state relations.
In the course of the article Archbp. Martin is called “Primate of Ireland”.
“But Father! But Father!”, some will say. “Isn’t the Archbishop of Armagh the Primate of Ireland? What gives?”
It seems the Archbishop of Dublin is, in fact, the Primate of Ireland while the Archbishop of Armagh is the Primate of ALL Ireland.
The dispute over who had the distinction was decided by Pope Innocent VI in 1353. The Archbishop of Armagh, Primate of All Ireland, has the seniority.
In England there is something similar. There was the Primate of England, the Archbishop of York, and the Primate of All England, the Archbishop of Canterbury.