To participate as a node relaying the block chain and transactions, a Pentium III is probably enough.
If you actually want to make bitcoin, you probably need to run several (ATI) GPUs or go with something more exotic like FPGAs/ASICs or "Big Iron". For GPUs I think the 4xxx series will work, the 5xxx series has best bang for you buck if you can find them, and the 6xxx series is currently being manufactured. For powering high-powered cards, you need a good power supply, preferably with Active Power Factor correction. You also may want to invest in a Watt meter like the kill-a-Watt and map your wiring before getting into mining in a big way.
My plan is to act as a node with token hashing power (drawing only 12 Watts from the wall with my sub-notebook) for about a year, and buy some Bitcoin as an investment. After a year, I may sell bitcoins to buy something like This
. I think that machine would do very well CPU mining; though likely drawing quite a bit of power. I would only turn on "threads" as network hashing power drops
(to protect the network and my investment in Bitcoin). Edit: that sever has hardware acceleration for SHA-256 (used in bitcoin mining (doesn't sound like per-core though))
Edit: the rated power draw is given as "cooling 7030 Btu/hr" (2060 Watts): you can't plug that into a standard home power outlet.
Edit2: This one
seems to be much better for "home" use: 9.4 Amps at 100 V, 4.7 amps at 200 V. Using those numbers the big one is about twice as power efficient though.