This helpful Euler diagram comes with the warning above:
The UK’s full name is “The United Kingdom of Great Britain and Northern Ireland”. People from the UK are called “British”. One British person is called a Briton. People from England are called English. People from Scotland are called Scottish. People from Wales are called Welsh. People from Northern Ireland are called Northern Irish. English, Scottish, Welsh and Northern Irish people are all British (whether they like it or not).
The ROI’s full name is “The Republic of Ireland” (if you are speaking English) or “Éire” (if you are speaking Irish). People from the ROI are called “Irish”. Irish people are not British. British people are not Irish. Irish people are not Northern Irish and Northern Irish people are not Irish. God help you if you forget this when you encounter an Irishman.
The ROI is not British. However, the “British Isles” include both the UK and ROI. Irish people hate this, but there is no consensus on what to call it instead. (May I humbly suggest “The British and Irish Isles”?)
And a hat tip to the mischievous Cormac!