The Taoiseach is the head of the Government. He or she is nominated by the Dáil (one of the Irish houses of Parliament) and is appointed by the President. The Taoiseach then nominates the other members of the Government. They are approved by the Dáil and appointed by the President. He may dismiss them if there is sufficient reason to do so. You can read this information on the current Taoiseach.
The Taoiseach acts as a channel of communication between the Government and the President. He or she advises the President about summoning and dissolving the Dáil. He or she presents Bills (or proposals for legislation) to the President for signature so that they can become Acts with legal effect. The Taoiseach also keeps the President generally informed on domestic and international policy matters.
As the leading public figure in the Government, the Taoiseach also acts as a spokesperson for the Government on major policy issues and chairs Cabinet meetings.
He or she takes an interest in the affairs of all the Departments of State and ensures that the plans of departments are co-ordinated. He or she makes sure that Cabinet decisions are carried out by the ministers and that the ministers fully understand the implications of different Government policies.
The Taoiseach also has a major role to play in our relations with foreign countries. He or she represents Ireland abroad and regularly meets with the other heads of state of the European Union to help shape European policy.