A graphic card is probably the most important part in a computer after the processor, mobo and RAM. A fast processor is no good if you combine it with a low end graphics card, especially if you are into a lot of gaming/multimedia creation.
The different catagories are:
Basic/ Internet PC -
This is a basic PC used for simple word processing, email, internet etc. Mainly used only for 2D applications. I recommend you go in for a nvidia TNT2 chipset based card with atleast 16MB VRAM. Or you could stick with the integrated solution of your mobo, if you have one with this feature.
Business PC -
The same as above. But if you use a projector/LCD display, get yourself a nvidia geforce 2MX with dual monitor/DVI support with 32MB VRAM.
Budget Home PC -
The home PC needs a higher graphics card. The reason here is that it will be used for all the above plus VCD/DVD viewing, gaming etc. I recommend a nvidia Geforce based solution with 32MB DDR VRAM.
Performance/Gaming PC -
This is when you have the best combination of hardware. It will have to support full 3D acceleration, support for high resolutions and extra features(TV in/out(VIVO), DVI support etc). Get a nvidia geforce3/geforce4 or ATI Radeon 8500 based cards with a bare minimum of 64MB DDR VRAM.
Design PC -
Ok so you are into high end 3D modelling and animation. Your best bet would be a nvidia Quadro based card or from 3Dlabs. These cards usually have drivers that are highly optimized for 3D apps, not to mention support for advanced features. But make no mistake even the geforce3/4 or the radeon are pretty good here.
Apart from these, all cards irrespective of the category should support:
DirectX
OpenGL
AGP 1x/2x/4x
24/32 bit true color