1) Avoid ATI at all costs! They have great hardware (Radeon series) but they couldn't write a decent device driver to save their corporate lives!
ATI's software is terrible, and it causes severe problems with their cards...
2) You cannot upgrade the RAM on a video card anymore. It's allways soldered directly to the card.
3) Any card that says 'TwinView' or 'Dual Head' means that it has ports for 2 monitors on the back, so you can hook 2 screens to the same card.
4) Allways buy nVidia cards. Their hardware is equal to or better than ATI (depends on which month it is), and their software is top notch. They support all major OSes (Mac, Linux, Windows 98-XPain), and I've never had a problem... (I've owned both ATI and nVidia. With ATI, the Linux XFree86 drivers (which were NOT written by ATI) were great, but the Windows ones that ATI wrote were TERRIBLE!)
Edited to add that I just looked at your system specs...
An Intel 350mhz won't have the hardware to take full advantage of a new card... It will work (and it's a good idea if you anticipate moving up to a new motherboard/RAM/CPU combo soon), but you should think about a new AMD based motherboard with DDR RAM soon.
A new mobo will fit in your existing case, so all you replace is the CPU, motherboard, and your system RAM. Stay away from Intel (they're slower over all, even though the clock rate is higher. Think stock Honda Civic vs stock 350 Trans Am - the Civic redlines at much higher RPMS, but is much slower), and get an Athlon XP.