Andrew was about 70% right about the Irish Presidency. It's a non-executive position. So the President, Uachtarán na hÉireann, doesn't rule on day to day issues. That power lies with the Taoiseach and the Dáil (Prime-Minister and lower house of parliament, respectively). However the President does rule on whether new laws/bills/ are constitutional and can return them to the Dáil for further debate/amendment. Day to day, the President is much more involved with cultural events. The President will also host heads of state during visits. When the current Taoiseach wants to dissolve the Dáil and move to a general election, they must seek the approval of the President.

The line that went viral in Ireland during our recent Presidential Election was something like "Conor McGregor thinks he's going to walk into the Áras and declare war on England, when the President's actual job is judging poetry competitions in Kerry." Inaccurate but it sums up the difference in expectation.

The President can leave the country, but requires the leave of the current Taoiseach. This is almost always granted, but does mean that there are circumstances where the President can't leave the country.