Statutory registration of marriages started in Ireland in 1845, births in 1864. Prior to that you have to rely on church records, very few of which are on-line. In addition some Church of Ireland records were lost in a fire in 1922 in Dublin (ironically they had been sent there for safe keeping).
So there aren’t many records for the 1700s and what little there is, in mostly not on-line. You need to go to PRONI in Belfast or one of the other repositories where they are held.
Griffiths is a substitute source for searching for ancestors in Ireland. The censuses from 1821 – 1891 have, in the main, all been destroyed. Some fragments have survived, including the 1831 census for most of Co Londonderry, but in general they are lost, so other substitute sources are used. The government conducted a survey of all land in Ireland in the period 1848 – 1864, so that accurate taxes could be levied. This record is known a Griffiths Primary Valuation and is on-line. It lists the heads of households where the rental value of a property was £5 a year or more. It is not as comprehensive a survey a census because obviously many people were excluded but in the absence of any other sources, it can help identify where ancestors lived. If you know the townland they came from you can often pinpoint their exact property. There is a facility on it for overlaying a modern map onto the contemporaneous map so making it easier to find a particular property today.http://www.askaboutireland.ie/griffith-valuation/index.xml