Prior to the introduction of civil registration in the mid-19th century (1845 for non-Roman Catholic marriages and 1864 for Roman Catholic births, marriages and deaths, and for non-RC births and deaths), the only place such information was recorded on a countrywide scale was in church parish registers.
Parish registers divide between RC and non-RC registers. If your ancestors were Roman Catholic, their baptism and marriage records will hopefully have been registered in the parish where the event took place. Note that an RC parish differs from a civil parish: it can have a different name and usually covers a larger area, taking in one or more civil parishes — an important distinction if you're researching on your own.
The challenge with RC registers is the piecemeal nature of their survival, and even where a register survives there's no guarantee the relevant entry is legible. Standardisation didn't arrive until the late 19th or early 20th century, so details recorded vary widely from parish to parish. Many entries are written in Latin (sometimes English, never Irish), so a grasp of common Latin terms helps — for example, a column headed “Sponsus” means “groom”, not “sponsor”. The limited range of 19th-century forenames adds confusion: a great many men were recorded as John/Ioannes and women as Mary/Maria, so a bride and a sponsor, or a groom and a sponsor, often shared a forename or surname.
Most RC registers date at the earliest to the early or middle decades of the 19th century, with an online cut-off around 1900; a few survive from the later 18th century, but rarely earlier. The registers remain the property of the parish and are generally not publicly held, though many are now online and fully searchable for a small fee.
Church of Ireland (Anglican) registers are mostly held centrally at the Representative Church Body Library in Dublin. Although often earlier in date than RC registers, fewer survive — many were lost in the 1922 fire at the Public Record Office.