Information from the British Embassy in Dublin:

"As the United Kingdom and Ireland forms a common travel area, it is not necessary for the nationals of those two countries to carry a passport when travelling between the two.

However, for their own administrative purposes, some airlines flying between the two countries require all their passengers to have some form of photographic identity with them when checking in.

Passengers are strongly advised to check with their airline, before flying, on exactly what their individual requirements are."