I would say the hard drive and speed of the dvd drive for copying and burning data. Also fast memory helps too and processor speed, especially if you plan on converting dvd movies to a different format like divx.
CPU is a big one when converting to Divx. I only did it once on my 1 GHz Athlon, the quality of the movie wasn't that great since I didn't know how to set it up optimally, and it still took something like 9 hours.
what most people dont relise is that when you convert from dvd to divx your comp is actually dsecompressing AND compressing. dvd aqre in mpg2 format which is compressed at up to 20:1 ratio from the original uncompressed video. before you can compress it to a divx avi it needs to be uncompressed. this is done on the fly as part of the diVX encoding. for good high quality divx you need high bandwidth and a powerful numbercrunching cpu. plenty of ram is good too. a MINIMUM of 512KB ideally
As the others have said - CPU and fast memory. Video apps tend to be multi-processor friendly so if you are really serious (i.e. spending all your time ripping) then a dual cpu system can save you some time. I often leave 3 or 4 FLASK sessions running overnight to get best utilisation.
For general use though a fast P4 or posssibly fast XP would be good. If you want the absolute fastest go for the P4 high end (3/3.2Ghz 800FSB). You'll pay a lot though. If you don't mind overclocking then everyone talks up the 2.4C P4 oc'd to 3/3.2 with good RAM and board. Hard to argue that.
-* <font color=red> !! S O L D !! </font color=red> *-
To the gentleman in the pink Tutu