Thewanderer has pretty much covered the Republic (and his photos are great as well).
NI Railways are effectively the last state-owned TOC in the UK, although the structure is slightly different; NIR's infrastructure is its own, there's no relationship to Network Rail.
As already mentioned, the Enterprise would be the only regularly scheduled loco-hauled service in Northern Ireland. The carriages used on the Enterprise are unique in Ireland, and were built by de Dietrich around the same time they were building the Eurostar. The interiors are very similar to the original Eurostar interiors, and if you squint you can see the similarity outside; only the Enterprise carriages don't have shared bogies.
There is one rake of British Mk. 2 carriages that were formerly used on the Gatwick Express. They were sold, re-gauged and brought over in the late nineties (IIRC?) and provided additional rush hour capacity on the Portadown - Belfast line before the full compliment of C3K trains arrived from CAF in Spain. I haven't seen the Gatwicks in passenger service for a long time, but I believe they're still operational.
There are 23 three-car C3K (3000 class) DMUs; all standard class, 90mph capable. Very comfortable, a giant leap forward for NIR.
Construction has started on the first of the next fleet to arrive, the C4K (4000 class). There's be about twenty of them, in three and four car configurations. Although they'll be similar to the C3K and interoperable, they will have many minor improvements based on the experiences of the C3K, so should be considered a distinct class.
There are 9 three-car 450 class DMUs, mainly working on the Larne line. These were built in the mid-1980s, using new mk. 3 derived bodyshells on recycled mk. 1 underframes and engines. The look a bit like a British class 150 or 210, but once you hear their unbelievable engine roar, you won't confuse them again. They've had their working life extended to last until the next tranche CAF built DMUs arrive, although continue to be pretty unreliable. Although my love for them has been eroded by having to use them as a form of transport, they're great fun to bash if you like that sort of thing
For an indication of their distinctive burble:
http://www.youtube.com/watch?v=XqBjaMZ2sik
The very last of the class 80 trains, built in the mid-seventies, can be found operating in four car configurations with two motor vehicles for reliability. As with most older UTA, CIE and NIR multiples, these are not fixed units. The driving vehicles could be chopped and changed with the trailers as was required, so additional mk. 2 carriages have been added removed from them throughout their working lives. You'll find the last class 80 (I think) on the Coleraine / Portrush shuttle, but they occasionally show up elsewhere.
There are three main routes in NI: Portadown - Belfast - Bangor is the busiest and best served corridor, with the largest number of commuters and is served by the C3K fleet. Belfast - Antrim - Ballymena - Coleraine / L'Derry is still due a lot of line improvements, but provides scenic and decent enough service to the Maiden City. It is, however, slower than taking the bus. If and when the work is fully completed, a 90 minute journey time from Belfast to Derry is on the cards, but that's a long way off.
Wikipedia has decent enough info on all the units and lines.
