People from The Republic of Ireland are Irish. Those from Scotland are Scottish and British, those from Wales are Welsh and British, and those from England are English and British.
There is a certain political sensitivity about people from Northern Ireland. They are British, in the sense that they are British citizens, but they may describe themselves in different terms depending on their religious and political beliefs. Many Unionists do not consider they are Irish (they are certainly not Irish citizens), but many Republicans do feel that they are Irish (though they are British citizens).