Most visitors to the country come to Dublin, and the city is an excellent choice for a short break, with many bars and museums to visit. The hotels around the Temple Bar area are the best to stay in, as they are the closest to the nightlife. It's worth noting that the Irish government banned smoking in bars and restaurants, so often you see fans of tobacco huddling out of the rain under awnings outside the bars.
Cork is another popular choice for a short break, and many visitors prefer the more laid back feel to busy Dublin.
Belfast in Northern Ireland is another increasingly popular city to visit, and has a charm unique from the Southern cities. It was an industrial city, and the scars of 'the troubles' are still present. As peace has come to Belfast, the city has become a much more pleasant place to visit, with glossy new bars replacing the smoky pubs. You'll hear politics talked about more than anywhere, but these days, talking is about as heated as Belfast gets.