It depends on the passport you hold. The fact that the U.K. and Ireland form a Common Travel Area without border control and that British and Irish citizens do not need a passport for travel by law [if not flying on Ryanair] does not exempt foreigners from holding a passport and the required visa for Ireland. You will need an Irish visa unless your country is listed on the attached website.
http://foreignaffairs.gov.ie/home/index.… ..
Edit: [based on your additional details]
Yes, as British citizen flying on Ryanair you will need a passport
http://www.ryanair.com/en/questions/what… ..
Visa are only required from non-EU citizens, but there are also some exemptions.
Flying on any other other company you would just need any type of government picture ID.
http://www.citizensinformation.ie/catego… ..

