If you are a British citizen you do not require a passport to travel to Dublin from the U.K. as it is part of the Common Travel Area. Similarly, I (as an Irish citizen) do not need a passport to travel to the U.K.
The Common Travel Area is a historical artifact and is due to the fact that following independence of twenty-six of Ireland's counties from the United Kingdom in 1922, no laws were passed requiring a passport for travel across the new border.
I live near the border between the Republic of Ireland and Northern Ireland (part of the U.K.) and I criss-cross the border regularly unhindered.
However, if you are flying, the airline will probably insist on photographic I.D., (usually a passport or driving licence)