On this map of the British Isles, it is easy to see the northern part of Ireland is part of the United Kingdom. Ireland was first invaded by the English a very long time ago (in 1170!!). After centuries of occupation, England finally left the isle, but kept one small part of Ireland: the "partition" (It took place in 1921).
The problem is: 1/ the majority of people in Northern Ireland are Protestants, and want London for capital (= today's situation). They don't want Dublin for capital (= they don't want to be part of a United Ireland).2/ the minority of people in Northern Ireland are Catholics and want Dublin for capital. They want to be part of a United Ireland.
Click here to read about the "Troubles".