Slightly less complex than that on the UK / Ireland thing.
The UK and Ireland have a common travel area. i.e. You can travel between the two countries without any form of ID. However, if you're traveling by air, you are required (by the Airline) to provide them with Photo ID, the exact requirements of which type vary by airline.
In general, a driver's license or passport for Ryanair
Aer Lingus : Driver's Licence, Passport, Work ID, Student ID etc.
others : varies
Same goes for an Irish or British domestic flight.
Basically what happened was the rest of the EU signed a common border agreement, the Schengen Agreement a few years ago which eliminated passport control between those countries that had signed up. The Schengen Agreement was also signed up to by Iceland and Norway who are not in the EU.
It also does not yet apply to new EU member states:
So, it only works in:
Belgium, France, Germany, Luxembourg, Netherlands, Portugal, spain, Italy, Austria, Greece, Denmark, Finland, (Iceland), (Norway) and Sweden. (Not in EU)
The timetable for New EU members to implement the agreement is unclear, but it's expected by March 29th 2008.
The UK opted out of parts of it and, because the Republic of Ireland shares a land border with the UK, we followed suit as otherwise we'd have had passport controls on our border with Northern Ireland which would have been very impractical as there are countless little roads that weave their way across the border, you wouldn't even know you'd crossed. There are even villages straddling it.
The two countries possibly have the most intertwined relationship in the EU. i.e. a UK citizen living in Ireland can pretty much do anything an Irish citizen can, including vote and fully take part in politics, as can an Irish citizen living in the UK. This is due to the historical fact that the Rep. of Ireland was part of the UK until 1921 etc etc. There's also a very deep and ever increasing level of cross border cooporation on law enforcement, infrastructure, etc etc etc.
The requirement is that UK and Irish Citizens (as we have no ID cards) must show a passport when entering the Shengen Area and Shengen Area citizens must show a passport or national ID card when entering Ireland or the UK.
However, the UK and Ireland did sign up to the other aspects of it, covering full police cooperation, the Schengen Information system (SIS) (allows us to see details of records for border security and law enforcement)
Other than that, there's nothing.
For a non EU national, bare in mind that you do need to carry a passport while in the EU. While there are no internal border controls, you can be asked to produce it.
There's also a Schengen visa which gives people rights to visit the entire Schengen zone. However, each country still has control of it's own work permits and long term visas / residency permits.