If you are a UK citizen, you need a passport to travel to other EU countries (with the exception of Ireland).
If you are an Irish citizen, you need a passport to travel to other EU countries (with the exception of the UK).
If you are from any other EU country, you need a passport to travel to the UK or Ireland.
If you are from one of the Schengen countries in the EU - Austria, Belgium, Denmark, Finland, France, Germany, Italy, Greece, Luxembourg, Netherlands, Portugal, Spain or Sweden, you can travel to any of the other countries in the group without a passport.
Don't know that the situation is if you are a citizen of one of the 'new' countries. I presume as they are not in Schengen, they will need a passport to travel to all other EU countries.
Phew! I'm sure I've made a mistake there somewhere.