The only thing you need really is a retail (these normally come with software included) DVD/RW drive in any sort of machine, its difficult to spec without a budget, do you want machine to be able to do anything else at all? or is it just for copying dvds?
AMD Athlon II X4 Quad Core 640 3.00GHz (Socket AM3) - Retail £82.24 (£69.99)
Corsair Dominator 4GB (2x2GB) DDR3 PC3-12800C9 1600MHz Dual Channel Kit (CMD4GX3M2A1600C9) £76.36 (£64.99)
Gigabyte GA-880GM-UD2H AMD 880G (Socket AM3) microATX DDR3 Motherboard £69.99 (£59.57)
Samsung SpinPoint F3 500GB SATA-II 16MB Cache - OEM (HD502HJ) £32.99 (£28.08)
Corsair CX 400W ATX Power Supply (CMPSU-400CXUK) £32.99 (£28.08)
Xigmatek Asgard Midi Tower Case - Black £26.99 (£22.97)
LiteOn IHAS524-32 24x DVD±RW SATA ReWriter with LabelTag - (Black) Retail £20.99 (£17.86)
Sub Total : £291.54
Shipping cost assumes delivery to UK Mainland with:
DPD Next Day Parcel
(This can be changed during checkout) Shipping : £11.75
VAT is being charged at 17.50% VAT : £53.08
Total : £356.37
You could add a few storage HDDs if you want to back DVDs up onto them too,
eg