It's an island. It's a country. It is not an island country, and the reason trips up more people than any other question in geography quizzes.
Ireland is an island. The Republic of Ireland is a country. The Republic of Ireland is not an island country. All three of those are true at once, which is why this argument comes up so often and why it rarely gets settled.
An island country is one with no land border. Ireland has one.
The island is shared. The Republic covers roughly five-sixths of it; the north-eastern corner is Northern Ireland, which is part of the United Kingdom. The border between them runs about 500 kilometres from Lough Foyle in the north-west to Carlingford Lough in the east. It crosses farmland, rivers and roads, and in a few well-documented cases it runs straight through individual buildings — there is a pub in County Cavan where you can order at the bar in one jurisdiction and drink at a table in another.
The usual objection is that the border doesn't feel like one. Since the Good Friday Agreement in 1998 there have been no checkpoints, no passport control and, for most people crossing it, no indication of anything at all beyond a change in road signs from kilometres to miles.
That invisibility was hard won and deliberately preserved. It is also precisely why the border became one of the most difficult problems in the Brexit negotiations: both sides wanted a land border that continued to behave as though it wasn't there, which is a genuinely hard thing to arrange between a country inside the EU customs union and one outside it.
But a border you can't see is still a border. Geography doesn't care how easy it is to cross.
If Ireland is disqualified by that border, so is the United Kingdom. The UK is usually pictured as Great Britain — England, Scotland and Wales on one island — but it also governs Northern Ireland, and that gives it a land border on Irish soil. Two countries, one border, neither of them an island country.
Great Britain itself is undivided between states. England, Scotland and Wales are nations within a single sovereign country, which is a different thing entirely from the island being split between two countries.
Once you apply it consistently, a surprising number of famous "island nations" fall away.
Haiti and the Dominican Republic divide Hispaniola between them, with a border running the length of the island. Neither qualifies, and each disqualifies the other.
Brunei sits on the northern coast of Borneo, an island it shares with Malaysia and Indonesia. It is small, coastal and surrounded by water on the side people picture — and it has a land border with Malaysia on every other side.
Papua New Guinea occupies the eastern half of New Guinea; Indonesia has the west. Timor-Leste occupies part of Timor; Indonesia has the rest. Both are archipelagic countries made up of many islands, and both are ruled out by a single land border on the largest one.
People get this wrong because they ask themselves the wrong question. "Is this country on an island?" is the intuitive test, and it fails constantly.
"Does this country touch another country by land?" is the test that actually works. It takes the same half-second to answer, it never depends on how big the island is or how friendly the border is, and it gets Ireland right first time.