The list of Commonwealth member states is here
http://www.thecommonwealth.org/Internal/142227 /members/
but you definitely won't find Ireland in it!
Ireland regained its independence from Britain in 1922, when the British Empire (out of which the Commonwealth was later formed) was still strong. It's unlikely that the Irish leaders, who had just fought a long a bloody campaign for freedom, would have wanted to have any sort of connection with the Empire. Further, even if they had wanted some sort of informal link to the Empire, this would have been impossible because, as a republic, they could not (at that time) have given allegiance to a body headed by the British monarch.
In 1949, India achieved independence from the Empire and it was becoming clear that the Empire was going to break up over the next few years. However, it was recognised that there were some benefits to retaining trade and cultural links between former member states of the Empire and so the Empire was 're-modelled' to become the British Commonwealth. (The word 'British' was, of course, later dropped from the title). This permitted the London Declaration which made it possible for a republic (such as India) to acknowledge the British monarch as Head of the Commonwealth, rather than as Emperor.
Although it's technically possible for a country with no colonial links to become a member of the Commonwealth, very few nations have sought to do so. (Mozambique and Cameroon are the only two so far). Ireland has, understandably, never felt the need to apply for membership.
Chris