You can find useful information regarding getting married in Ireland from the appropriate Irish government authorities on the following websites:
Health Service Executive: http://www.hse.ie/eng/services/list/1/bdm/marriagesinireland/
General Register Office: https://www.welfare.ie/en/Pages/Getting_Married.aspx
In order to get married in Ireland, you may be required to provide your U.S. birth certificate with an Apostille for the Irish Authorities. The U.S. Embassy Dublin cannot complete Apostilles on U.S. issued birth certificates for the purpose of getting married in Ireland. The Secretary of State offices in each state have control of vital record documents, please contact your State’s Secretary of State office for instructions on how to get an Apostille.
You may also be required to provide a Civil Letter of Freedom to Marry for the Irish Authorities. Unfortunately, the U.S. government does not have centralized registry of vital records and we are, therefore, unable to provide you with a Freedom to Marry certificate. You may be able to obtain a Freedom to Marry certificate from your local county in the United States, but you should check with the Irish authority who will be marrying you whether this will be acceptable.
Alternatively, the Irish Commissioner of Oaths could notarize a statement which states that you are free to marry.
Another option, once you confirm with the authorities you are submitting this to that they will accept this document, is to get an Affidavit of Freedom to Marry completed with the U.S. Embassy: