To teach in Ireland, you generally need a postgraduate qualification. This is true for teachers in primary and secondary schools, and of course, at third-level institutions.
For primary teaching, you need a Bachelor of Education (BEd), plus a postgraduate qualification, such as the Professional Master of Education in Primary Education. For secondary teaching, you'll need the Professional Master of Education (Post Primary Education). If you have a Level 8 degree, you can do a postgraduate conversion course to qualify as an education professional.