This is one of the most frequently asked questions on AB but the questioners never seem to mention which airline (or ferry company) they plan to travel with!
The UK and the Republic of Ireland operate an 'open border' policy so, as far as the immigration authorities are concerned, no ID is required. However, each carrier will have their own policy on acceptable forms of ID for security purposes. (So you need to check with the relevant carrier). If you're travelling with Ryanair (and you were born within the European Economic Area), then an EEA driving licence, with photocard, is acceptable ID on flights between the Republic and the UK:
http://www.ryanair.com/site/EN/faqs.php?sect=p id&quest=photoid
Chris