Final answer:
The large island off the coast of mainland Europe referred to in the question is home to the United Kingdom, which makes up the majority of the island of Great Britain in the British Isles.
Step-by-step explanation:
The large island just off the coast of mainland Europe is home to the country known as the United Kingdom (UK).
The UK is part of the British Isles, an archipelago separated from the European mainland by the English Channel.
The British Isles consist of two countries; the Republic of Ireland and Great Britain. Great Britain itself includes England, Scotland, and Wales.
This distinction is important to note because, while Ireland is indeed a large island off the coast of Europe, the question specifically refers to a single country occupying the island, which is the UK in the case of Great Britain.