Not quite sure what you are after but I'm thinking probably the following (for a wired connection - I'll do wireless in a minute):
1.Network card (by someone like Belkin or Netgear) - shouldn't cost more than about £15 - open up your computer an click it into a slot on the motherboard. If there aren't any free slots, take the old modem out - you won't need it any more.
2. Network cable - I'd advise you to get this from a local, independent PC shop simply because most will cut you a custom length and will charge you much less than you'll have to pay for what is effectively the same stuff if you buy it from somewhere like PC World (do they have them in Ireland?).
3. Router - The most expensive part - will probably cost about £50. D-Link are good although I've just changed my router to a Netgear which seems to be doing the job nicely.
I wouldn't necessarily recommend wireless, but if you want it then you can either get a PCI card, which fits exactly the same as I said for the wired above, or an antenna thing which plugs into the USB (much easier but not necessarily as reliable). Obviously you won't need wire but you will need a wireless router - the type you'll need will depend on what you'll be using it for.
Wired is probably slightly cheaper and easier to set up and more reliable - wireless is tidier and has the obvious advantage that you connect any (wireless enabled) computer to it anywhere in the house.
If you go for wireless make sure you secure it properly or anyone can steal it.
EDIT: **** - just realised how long ago you posted this - this place is so ******* slow now.