Most people tend to visit Ireland in July and August, so it will be really busy around that time. Good times to visit Ireland would be in May, June, or September. There are many different kinds of festivals that go on in October, so if you are interested in that, October would be a great time!
Ireland is beautiful in the spring. I actually saw lambs skipping over green hills, just like the postcards. Also, if you're a James Joyce fan ? or aspire to be one ? you'll want to be in Dublin on Bloomsday, June 16th.