Whether you are trying to apply for citizenship in America or trying to obtain a passport to leave Ireland and go on vacation, you are going to need a copy of your birth certificate. If yours has been accidentally misplaced or destroyed, you can get a copy that will be accepted by authorities as if it were the original.
Check out groireland.ie and obtain a copy of a birth certificate request form. It's available in a PDF file.
Fill out the form. You should know the name of the hospital you were born in and its location, and the towns your parents hail from. This will narrow the search for your records. Check the "long form" version of your birth certificate, which is recognized by other state offices.
Pay by cash, check or credit card. Because the exchange rate is constantly in flux, it's probably easiest to pay with a credit card. They'll do the currency conversion. If you happen to be in Ireland, you can pay in Euros.
Enclose your payment option and completed form in an envelope and send it to: Civil Registration Office, Office of the Registrar General, Government Offices, Convent Road, Roscommon, Co. Roscommon, Ireland.