The President is elected by the people of Ireland for a seven-year term and has a largely ceremonial role. The Taoiseach is appointed by the President on the nomination of the Dáil Éireann (lower house of the Oireachtas, the Irish Parliament).
The Oireachtas is composed of the Dáil Éireann, the Seanad Éireann (upper house), and the President. The Dáil is elected by the people of Ireland using proportional representation, with 160 members. The Seanad is composed of 60 members: 43 are elected by the Dáil, 11 by the National University of Ireland and the University of Dublin, and 6 are appointed by the Taoiseach.
The government is formed by the party or parties that hold a majority in the Dáil. The Taoiseach is the head of government and chairs the Cabinet, which is composed of the ministers of state. The Cabinet is responsible for the day-to-day administration of the country.
The Oireachtas has the power to pass laws, impose taxes, and approve the government's budget. The President has the power to sign or veto legislation, but the Dáil can override a presidential veto with a two-thirds majority vote.
The Irish system of government is based on the principle of parliamentary sovereignty, which means that the Oireachtas has the final say on all matters of law. However, the President does have some reserve powers, such as the power to refer certain bills to the Supreme Court for review.
The Irish government is also subject to the rule of law, which means that it must operate within the framework of the Constitution of Ireland. The Constitution can be amended by a referendum, which requires the approval of the people of Ireland.