Public health care is free at the point of use, meaning you don't have to cough up several thousand (or several HUNDRED thousand) for a relatively-simple operation like you do in the US, but you pay through your taxes.
However, like everywhere else, private health care is not free (hence the term "private").
As a result, taxes in Ireland (and other countries with similar systems) are a bit higher than in the US, but, on the other hand, medical bankruptcy is virtually unheard of here and (perhaps surprisingly to some people) the quality of treatment in the public system is often the same as, or even better than, that available in the private system. In fact, the only real advantage of "going private", as they say, is speed of treatment but, since emergency cases get priority in the public system anyway, it usually amounts to little more than buying your way out of impatience (and I say that as someone who has, and still does, use(d) both systems).