What do I need to use two 24" monitors on one pc?

Heavy stuff. :) Well done!
I think it will work, but only if your card has two VGA ports, what you of course already have checked out…

First, is that video card “dual headed” does it have either two VGA connectors or two DVI connectors or a dongle which splits a connector on the video card to either dual VGA or dual DVI?

Secondly, I would highly recommend if you are going to push that high of resolution that you stay away from VGA all together. VGA is anaglog and you will have trouble with it. DVI is digital and has a larger pipe through which to push video data to the monitor. Both your card and your monitor have to support DVI for it to work.

Here is what VGA looks like:
http://en.wikipedia.org/wiki/Vga

Here is what DVI looks like:
http://en.wikipedia.org/wiki/Dvi

What model of Dell monitor are you looking at?

The card has two DVI connectors, and the Dell monitors are both the UltraSharp 2407WFP. I wonder if the video card is going to be able to output 1920×1200 on each port, totalling a resolution of 3840×1200. Thanks for your help!

You should have no trouble at all driving that resolution. I would suggest that you get one of the many multimonitor management apps. I happen to use UltraMon but there are many others. In fact there was a good article on LifeHacker recently about multimonitors.

http://lifehacker.com/software/geek-to-live/make-the-most-of-your-dual-monitors-317479.php

Good Luck.