Why is part of Ireland owned out by the UK?

I just realized that. I am aware that Ireland used to be part of the UK about over a century ago but why is a part of Ireland still part of the UK? I don't understand why that is. They should have claimed independence for all of Ireland and not just east, west and south. Its just uneven in my oppinion. If you look up the UK anywhere you will hear "United Kingdom of Great Britain and Northern Ireland" which is so half azz. lol