Well like Scotland who in 1706-1707 signed the Treaty of the Unions and the Act of Unions there was a similar agreement with Ireland in the Act of Union 1800 that officially brought Ireland on board with the rest of the United Kingdoms, disputes within Ireland led to the partition of Ireland in 1921 and in 1922 The Irish free state was given dominion status while the North remained part of the UK and it's title became United Kingdom of Great Britain and Northern Ireland
So i would say NI is part of the United Kingdom of Great Britain
Some people have the view like Trojan that is an occupied part of Ireland as a whole which is another ball game entirely and some people would say the very same about Scotland,Wales being occupied very complex situation as i think has been said on here where does it end.
Being part of the United Kingdoms in modern times does'nt seem to have been such a bad thing as it is quite a reasonable place to reside compared to a few countries.
But other people may disagree with this depending on there train of thought