Ok, Lots of confusion here.
The UK is NOT a Country. The UK is a Sovereign State, consisting of four countries:
England
Northern Ireland (Not North Ireland as a couple of people are stating)
Scotland
Wales
Likewise, Great Britain is not a country:
England, Scotland and Wales all lie on an Island called Great Britain (Or Britain for Short) - So therefore, Britain is an Island.
Northern Ireland and The Republic of Ireland lie on the island of Ireland.
The Republic of Ireland, as its name suggests, is a Republic, and has nothing to do with Britain or the UK (Different laws, Currency, Government etc)
EDIT:
Lord Oxford: The Capital of Northern Ireland is Belfast.
Dublin is the Capital of the Republic of Ireland.
Also - Wales more popularly recognised as a Country, and not a Principality.