As far as I understand these things, it really depends on your motherboard and also the type of card you wish to install. Some cards are PCI and others are AGP. More powerful cards will presumably initially require a more powerful CPU.
The more powerful cards normally have inbuilt processors which 'process' the image data allowing the main CPU to 'concentrate' on other tasks after it has passed the data to the card.
The more powerful the card (128MB or 256MB), the more computer RAM you will need to support that card.
Each card has own requirements which are normally listed on the box.
The biggest thing to be concerned with is not the processor itself but the motherboard. You need to find the manual for the motherboard (or download it from the internet) to see what type of slot you have. For a P3, you probably have an AGP slot, although some less expensive motherboards skipped this - often the small footprint PCs from major manufacturers.
You should know what the speed of the AGP slot is - the orginal AGP specification is 1x, but today you can get cards that run to 8x - no point in getting a super fast 8x card for a 1x slot. If you don't have an AGP slot(and you can have AGP video on the motherboard without having an AGP slot BTW) you can get PCI card, which are not as fast but will do the job.
Are you wanting to run games on this PC? You probably want to look at a balance of a good card and enough RAM. There is no point in getting a killer $300 video card for a P3. If you are gaming, at least 512MB of RAM is a good idea. On a brand new PC, I would suggest 1 GB of RAM. I'm currently using a 128MB video card on a P4 1.6.
There are 2 main chipsets for gamers - ATI and nVidia. Both have pros and cons, in some cases one is better for one game, the other another game. I would look at www.tomshardware.com for more information.
For an older PC like yours, I would suggest thinking about finding a refurbished card, unless you are planning on building a new system soon and moving your video card.
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compatibility ov video cards w/ a proc??