Answer:
Ireland is in fact a country and an island. Ireland and Éire are the official names for the country. Éire is the word you will see on Irish coins for example, and it means Ireland in English. It is located to the West of Great Britain. Great Britain comprises of three regions - England (which takes up the majority of the island), Wales (in the South-West of the island) and Scotland (in the North of the island).
The island of Ireland comprises of the Republic of Ireland and Northern Ireland. The Republic of Ireland takes up five-sixths of the island of Ireland. Northern Ireland takes up one-sixth of Ireland in the north-east of the island.
Northern Ireland is a part of the country known as the United Kingdom of Great Britain and Northern Ireland often abbreviated to United Kingdom or the UK. Northern Ireland is under the rule of the British monarchy and British Prime Minister. Like England, Scotland and Wales; it has a certain amount of government control (in the same way as US states) but are overall run by the Prime Minister of the UK.
The Republic of Ireland (which is generally just called Ireland) is a separate nation run by its own sovereign government. It has its own currency, own education, its own health service etc. and is a member of the United Nations. The United Kingdom is also a member of the United Nations. Ireland (Republic of Ireland) is not under the rule of the UK.
People born in Northern Ireland do have the unique choice of being able to choose whether they are Irish or British. Although technically citizens of the United Kingdom, they are legally allowed to refer to themselves as Irish if they like, generally due to political, cultural or religious reasons.