As far as the immigration authorities are concerned, UK and Irish citizens do not require a passport to travel between the two countries. (There's been an open border policy ever since Ireland regained its independence from the UK).
All airlines require some form of ID. The exact rules vary between airlines but all airlines flying between the UK and Ireland accept a photo-card style driving licence. Ryanair only accept passports, driving licences and (for nationals of those countries which have them) national ID cards. Other airlines, such as easyJet, are far more flexible and accept any photographic form of ID, such as a bus pass.
Chris