If you're travelling from the UK to Ireland, the Irish immigration authorities don't require you to take a passport. (Ireland and the UK operate an 'open border' policy).
However, if you're travelling by air, the airline will require you to produce suitable ID before allowing you to board the aircraft. Each airline has their own policy as to the type of ID which they'll accept. If you're not sure, either check the airline's website or post here again, stating which carrier you'll be travelling with. (Ryanair, for example, will only accept a passport or photo-style driving licence. See here for full details:
http://www.ryanair.com/site/EN/faqs.php?sect=p id&quest=photoid )
Chris