* The UK is the "United Kingdom of Great Britain and Northern Ireland". Great Britain is the largest island in its archipelago (the British Isles); Northern Ireland is a region situated on the island of Ireland (another island in that archipelago).
* Another region of the island of Ireland is (the bulk of) the sovereign state the Republic of Ireland, a country which is part of the EU. The Republic of Ireland is the country where many companies headquarter in the EU for tax efficiency.
* Alongside Northern Ireland, the other three countries which make up the UK are England, Wales, and Scotland. Those three countries are all primarily located on Great Britain. Some other islands in the archipelago are not part of the UK at all (e.g. the Isle of Man); some other islands in the archipelago are part of one of England, Wales, Scotland, or Northern Ireland.
To confuse the island of Ireland with the United Kingdom, or to identify the Republic of Ireland as part of the UK, is deeply offensive to some (many?) Irish people. It is also a sore point to certain people across the island of Ireland that Northern Ireland is part of the UK, and that the island of Ireland is not a unified state.
Confusingly enough, citizens of the UK are almost always referred to as "British citizens" whatever their relation to the island of Great Britain; this is a strange piece of terminology because if "British" here means "Great British" then it appears to exclude Northern Ireland, while if it means "of the British archipelago" then it appears to include (for example) the Republic of Ireland.
