Just that.
Ireland is a sovereign state, not part of the U.K. It hasn’t been part of the U.K. since 1922.
Northern Ireland is part of the United Kingdom. It’s not part of Great Britain though.
Some people refer to Ireland as the Republic of Ireland, partly to differentiate it from Northern Ireland, but the legal name of the state is Ireland.
While I’m at it, the term “British Isles” is controversial and the product of colonialist geography (geography is not a value-free discipline obviously). The British Gov itself apparently has internal guidance not to use it. In joint documents the British and Irish Govs use the euphemistic phrase “these islands”, and other options include Western Atlantic Archipelago or Islands Of the North Atlantic.
There’s no such country as “Southern Ireland”. And “Eire” is also incorrect, unless you’re in the habit of referring to Germany as Deutschland or Spain as España.