Statutory records of birth, death & marriage records started in Ireland in 1864 (1845 for Protestant and Non-Conformist marriages). Prior to that you are heavily dependant on church records, where they exist. In general, there are few records prior to 1800.
Ideally, to trace an ancestor in Ireland through church records you need to know their religion, townland and parish. To find that information if the person died overseas, the usual sources are naturalisation records, gravestones, family bibles, wills and obituaries.
Some Irish BDM records are searchable on line on the LDS pilot site, but some records are still held by the individual church (some declined to allow their records to be copied) and so this website is not the complete picture:
http://search.labs.familysearch.org/recordsearch/start.html#p=allCollections&r=1For Northern Ireland, the Public Record Office (PRONI) has copies of many of the parish registers, and has a list of those held elsewhere.
www.proni.gov.uk/The records themselves are not on line and you need to search them in person. PRONI is temporarily closed until April 2011 but there are temporary arrangements to view this data at Cregagh Library, Belfast. See PRONI website for details.
You say that your ancestors may come from Armoy. There is one McCloy family in the 1901 census in Armoy who are RC:
www.census.nationalarchives.ie/pages/1901/Antrim/Armoy/Breen/920894/Elwyn
