It's quite complicated, the Irish Republic was proclaimed in 1916, the Irish Free State (Soarstat Eireann) was the result of the 1921 treaty (Anglo-Irish agreement) which also partitioned Ireland. The 1921agreement was used as a basis for further removal of British rule, as Ireland was still a Dominion of the British Empire. The 1937 Constitution, ratified by popular vote, essentially caused the Free State to cease existence, making Ireland a Republic in all but name, the name of the country from that point was Ireland (Eire in Irish), I would caution you against using the name Eire, particularly when speaking English, as there is a bit of history there with regards to it being used as a term to de-legitimise the new Irish state in certain British establishment circles.

The Republic of Ireland Act of 1948 was essentially just a copper fastening of the 1937 Constitution, officially removing the last vestiges of British association with Irish government. As you can see, it was a drawn out process, which is why Easter 1916 is commemorated, there will be commemorations for the next few years, the first Dail of 1919, and the Treaty, however, they are likely to be more low key, particularly in relation to the Treaty, which divided Ireland...