I currently have a dynamic website that runs on php,mysql,sendmail
etc.  Opensource programs running on a Linux dedicated server. I'm
looking to buy a desktop computer within the next few weeks that will
be able to hold daily back ups on the hard drive. I'm planning on
programming daily CRON JOBS from my server to the new computer that
will be placed at my home office. The info that I will be backing up
is similar to craigslist.com info.  Strings of postings and things of
this nature.
QUESTIONS:
Do I need any certain programs for this? If so which ones are easy to
use and not so expensive?  What is the best way to go about this?
What should I be looking for when searching for a desktop computer. 
What kind of cost are we talking about. This doesn't need to be robust
just doable for now. What is the most important factor when
determining what computer I should buy?

To back up, you only need tar, unless you want something more... gui.

im intrested in the nature of your question... you know what a CRON
job is but you cant find the right puter for your work?  anyways any
mother board with 800mhz fsb and ddr will do, if you want a fast hdd
then go for a SATA capable motherboard and get a SATA drive, it dosent
matter what chipset you choose (AMD, Intel, etc.) and also what OS do
you plan on using for this office computer? if windows go with
windowsNT if linux go with redhat. anyways you can easyly set it up so
you can compress the data before you send to your office comp
(recomended) or you can compress it when it gets there... work with
batch files or "bash" files lol,,, you dont need any special software.