Just for Mining:
AMD Sempron 140
ASRock 880G Extreme3
2 GIGs of DDR3 Ram
a 4 GB USB Stick or the Ubuntu Operating System
up to Three ATI Radeon HD5850 Graphic Cards (the more Cards, the More Hashingpower you have)
And a Power Supply with at least 80+ silver Certificate (gold is better) with around ((Number of Graphic cards*150 Watts)+250W). So für One Card, a 400W power supply is sufficient
For other uses (Gaming) too: Replace the Processor with a AM3 Quadcore processor, Add 2, or better 6 GIGS of Ram and buy a harddrive and a DVD RAM Drive and you are fine.