Let's separate 'the authorities' from 'the carriers':
The UK and Ireland have an agreement that provides an 'open border' between the two countries for citizens of either country (who were born in the respective countries). So, as far as the immigration authorities (in both Ireland and the UK) are concerned, you do NOT need a passport to travel to and from Ireland. (That assumes that you're a British citizen, who was born here).
However airlines are required to carry out ID checks for security reasons and ferry companies are entitled to do so.
Ryanair has opted for a strict 'passports only' rule on its flight between the UK and Ireland:
http://www.ryanair.com/en/terms-and-conditions/regulations-traveldocumentation/
Aer Lingus is far more flexible. You can use (among other things) a photo-card driving licence or a bus pass:
http://www.aerlingus.com/travelinformation/knowbeforeyoufly/visaspassports/#d.en.4568
BA also only requires 'some form of photographic identification, such as a driving licence':
http://www.britishairways.com/en-gb/information/passports-visas-and-api#passports
CityJet's website mentions that Irish citizens can use a driving licence but, oddly, doesn't mention the same thing for UK citizens. (Phone them to check if you're considering using their flights):
http://www.cityjet.com/flying-with-us/terms-and-conditions/
There is no ID required to enter the Republic of Ireland from Northern Ireland, whether by bus, car or rail. (Once again, that applies only to British and Irish citizens, born within the respective countries). For flying to Northern Ireland you'll need to comply with the rules of the relevant airline. (easyJet used to accept almost anything with a photo on it, such as a bus pass, but their website no longer seems to mention it):
http://www.easyjet.com/en/help/preparing-to-fly/travel-documentation
Ferry operators require passengers to carry ID (but probably won't ask to see it anyway). For example, Irish Ferries accept driving licences, bank cards, utility bills, etc:
http://www.irishferries.com/uk-en/faq/passports-identification/
P&O just specify 'photographic identification':
http://www.poferries.com/eu/terms-and-conditions
Stena Line states 'a driving licence, citizenship card or utility bill will usually suffice':
http://www.stenaline.co.uk/FAQs/passports-and-visas/im-a-british-irish-citizen-do-i-need-a-passport-to-travel
Hoping that covers it all!