Ireland, along with the UK, is a member of the Common Travel Area. British nationals travelling from the UK don’t need a passport to visit Ireland. However, Irish immigration officers will check the ID of all passengers arriving by air from the UK and may ask for proof of nationality, particularly if you were born outside the UK. You are therefore advised to take your British passport with you.
For more information about the types of ID you might be asked to present, see the website of the Ireland Citizens Information Board.
Most airlines and other transport providers won’t carry passengers to and from Ireland unless they have seen satisfactory photographic ID. Check with your carrier before travelling.