The Irish immigration authorities don't require any form of identification for passengers arriving by air from the UK.
The airline which you're travelling with, however, will require you to produce valid photo identification before allowing you to board. Different airlines may have different requirements. If you're travelling with Ryanair, the only valid form of ID (other than a passport) for UK citizens is a photo-style driving licence. (All adults in the group must have a passport or driving licence. Accompanying children do not require photo-id).
Confirmation of Ryanair's requirements is here:
http://www.ryanair.com/site/EN/faqs.php?sect=pid&que st=photoid
If you're travelling with another airline, you'll need to consult their website (or post again and I'll find the information for you).
Chris