Greg, the first name for the 26 counties was the Irish Free State. This name was established by the 1921 treaty, which gave Ireland some measure of independence ffom the UK but within what was then the British Commonwealth (later just "the Commonwealth").In 1937 the Irish Free State adopted a new constitution (the one still in effect now, although there have been significant amendments). At the same time the country gave itself a new name: Eire.
On Easter Monday 1949 Eire left the commonwealth. At the same time it dropped the name Eire and became the Republic of Ireland.