As Mushroom25 states, British citizens do not require a passport to enter the Republic of Ireland.
However, you require suitable ID before an airline will issue you with a boarding card. Each company has its own rules. Ryanair, for example, will only accept a passport or a driving licence (with photo). Aer Lingus will accept almost any form of photo-ID, including a bus pass or a work's ID card.
Check the website of the carrier you intend to travel with (or post again on AB) to find out the rules which will apply to you. (Neither Ryanair nor Aer Lingus require photo ID for children, under 16, travelling with an adult to Ireland. Other carriers probably have similar policies).
A first passport for an adult currently costs �72. the fee for a first child passport is �46:
http://www.passport.gov.uk/fees.asp
(Use the links, on the left of that page, for further information about applying for passports).
Chris