Under the Constitution of Ireland, the Taoiseach is . They are then formally appointed to office by the President, who is required to appoint whomever the Dáil designates, without the option of declining to make the appointment. For this reason, the Taoiseach may, informally, be said to have been "elected" by Dáil Éireann.
It seems to me to be mere make-believe to try to incorporate a word like "Taoiseach" in the English language. It would be pronounced wrongly by 99 percent of the people. I have already ascertained it is a very difficult word to pronounce correctly. That being so, even for the sake of the dignity of the Irish language, it would be more sensible that when speaking English we should be allowed to refer to the gentleman in question as the Prime Minister... It is just one more example of the sort of things that are being done here as if for the purpose of putting off the people in the North. No useful purpose of any kind can be served by compelling us, when speaking English, to refer to An Taoiseach rather than to the Prime Minister.
Since the Taoiseach is the head of government, and may remove ministers at will, many of the powers specified, in law or the constitution, to be exercised by the government as a collective body, are in reality at the will of the Taoiseach. The Government almost always backs the Taoiseach in major decisions, and in many cases often merely formalizes that decision at a subsequent meeting after it has already been announced. Nevertheless, the need for collective decision making on paper acts as a safeguard against an unwise decision made by the Taoiseach.
Some biographies of former Taoisigh and presidents of the Executive Council: