Ireland is a member of the European Union.  Different rules apply on whether your journey originated in another EU state or outside the Union. 

If your journey commenced in another EU country, you can bring in goods on which tax has been paid in the other EU country provided the items are for your own personal use and do not contain prohibited items such as illegal drugs, firearms or meat or meat products.  If your journey commenced in another EU country you should exit through the Blue channel at the airport of arrival.  Since the UK is also part of the EU these rules also apply to travellers entering the Republic by land from Northern Ireland and there are no formalities at the border crossings.

If your journey commenced outside the EU (even if you transited in an EU country) the the following limits are applied : 200 cigarettes, 1 litre of spirits, 2 litres of wine and €175 of other goods. A full list is available on the Irish Reveune Commissioners website.  If your journey commenced outside the EU you should exit through the Green or Red channels at the airport.