What is the area of Ireland that is independent of Great Britain called?

Currently, it is the 'Republic of Ireland'.

When it 'broke away, from Great Britain, in 1922, it was named the 'Irish Free State'.

Then in 1937 it was renamed 'Eire'. The Irish (Erse) language name for Ireland.,

In 1949 it renamed to the modern current name of ' Republic of Ireland'.

Currently, it has a president, and a parliament (Dail) ( pronounced 'doyle/doil), based in its capital city of 'Dublin'.

Historically the island of Ireland was divided in to four provinces. viz., Ulster(North), Leinster(East), Munster(South) and Connaught/Connacht(West). Each province is divided in counties. On separation in 1922, when the Irish Free State was set up , the international boundary between the Irish Free State and the U.K separated the nine counties of Ulster. into six counties of Northern Ireland(UK), and three counties remaining in the Republic of Ireland. The other three provinces all remain in the Republic of Ireland.

The interntional border between the Republic of Ireland and Northern Ireland(UK) is a very open border. However, The Republic of Ireland is NOT a member of the British Commonwealth of nations.