Both the Republic Ireland and Northern Ireland observe the following holidays:

March 17th (St. Patrick's Day)
The Monday following Easter Sunday
Christmas Day
December 26th (St. Stephens Day (Republic) - Boxing Day (Northern Ireland).
Good Friday (although not officially recognized as a National Holiday)

The following are bank/national holidays in The Republic:

The first Monday in May
The first Monday in June
The first Monday in August
The last Monday in October.

The following are bank/national holidays in Northern Ireland:

The first Monday in May
The last Monday in May
July 12th (Orange Day)
The last Monday in August.