The Irish Parliament was subject to an Irish executive, presided over by the English/British selected Lord Lieutenant of Ireland (previously called the 'Lord Deputy'), which was ultimately answerable not to it but to the English/British Government in London.
Over the centuries, the Irish parliament met in a number of locations both inside and outside Dublin. Among its most famous meeting places were Dublin Castle, the Bluecoat School, Chichester House and its final permanent home, the Irish Houses of Parliament in College Green, also sometimes called the Irish Parliament House. It is now generally called the Bank of Ireland, an institution who took ownership of the building in 1804 and used it as its headquarters until the 1970s, when a new headquarters was built. The former seat of parliament remains a branch of the bank.
The single chamber Dáil Éireann (Assembly of Ireland), also known as the First Dáil formed by Irish MPs elected to the British House of Commons, who assembled in Dublin in January 1919; (following an election in 1921, the new membership called themselves the Second Dáil. All subsequent Dála (plural of Dáil) count them number back to the First Dáil)
Dáil Éireann was governed by a temporary constitution, called the Dáil Constitution, which created a prime minister called the President of Dáil Éireann (also known as Príomh Áire) and a ministry called the Áireacht. The prime ministerial office was updated to a headship of state, called President of the Republic, in August 1921.
This Dáil met in two locations, in the Round Room of the Mansion House, and later in the then University College Dublin campus in Earslfort Terrace in Dublin, where its famously ratified the Treaty in December1921.
In 1922, a government theoretically answerable to the House of Commons of Southern Ireland, called the Provisional Government, was created under Michael Collins.
The House of Commons of Southern Ireland officially was based in the Royal College of Science in Dublin, now the Irish Government Buildings. However the House of Commons only met on a handful of occasions, primarily to ratify the Treaty in January 1922 and confirm Michael Collins as head of the Provisional Government.
The bicameral Oireachtas Éireann, made up of the King and two chambers, Dáil Éireann' (The Chamber of Deputies) and Seanad Éireann (Senate of Ireland), created by the Irish Free State's 1922 Constitution (The Seanad and the King's role in the Oireachtas were both abolished by Constitutional Amendment in 1936)
The executive answerable to Dáil Éireann was called the Executive Council and was presided over by a prime minister called the President of the Executive Council.
The Provisional Government under W.T. Cosgrave hired the use of the main lecture theatre of the Royal Dublin Society in its headquarters in Leinster House, a formal ducal palace, to enable a formal state opening of the new two chamber Oireachtas Éireann of the new Irish Free State and the delivery of the speech from the throne by the new Governor-General of the Irish Free State, Tim Healy in December 1922. In 1924, the new Free State under Cosgrave bought Leinster House as a temporary seat of parliament, pending the erection or conversion of an alternative. One major contender for the location was the Royal Hospital, Kilmainham, a former soldiers' home that ultimately became a modern art gallery.
The executive answerable to Dáil Éireann is called the Government and is presided over by a prime minister called the Taoiseach.
Though plans were periodically discussed for the erection of a new parliament building (a site was even considered in the Phoenix Park), parliament remained in Leinster House, to which additional offices were added in the 1950s and most recently in the year 2000.
