Just to add to the mix:
Ulster <> Northern Ireland.
The island of Ireland has 32 counties (26 in the Rep., 6 in NI).
Also contains 4 provinces. These 4 are Leinster, Connacht, Munster and Ulster. (Might recognise some rugby teams named after these!).
Ulster is made up of the 6 counties of NI - Fermanagh, Antrim, Tyrone, Down, Armagh + Derry/Londonderry (last one's name depending on who you are), *plus* 3 counties Donegal, Cavan and Monaghan from the Republic.
So to say Ulster meaning NI, while it is done, (e.g. "Ulster says No") isn't strictly correct.
If you want to further wonder about all this look at sports:
Soccer - national teams for England, Scotland, Wales, NI, ROI
Rugby - national teams for England, Scotland, Wales one team "Ireland" - which is ROI+NI
Athletics - Team GB (England, Scotland, Wales + NI - should probably be called Team UK) and Team Ireland (ROI only)
Why all the differences??