XNA Game Studio (Express) 1.0, 1.0 Refresh and 2.0 require a DirectX 9 graphics card capable of Shader Model 1.1. In addition many of the creators club samples and starter kits will require a Shader Model 2.0 capable card.

XNA Wiki has a list of cards that work and do not work.

If your card is not on that list then there are many places to check the capabilities of your card:

http://en.wikipedia.org/wiki/Comparison_of_ATI_Graphics_Processing_Units
http://en.wikipedia.org/wiki/Comparison_of_NVIDIA_Graphics_Processing_Units

http://en.wikipedia.org/wiki/Pixel_shader

Make sure the card supports at least DirectX 9 and shader model 1.1. If your card isn't on the list, chances are you need to get a new one, but a quick Google should be able to verify this.

If you are prepared to run a 3rd party application (Microsoft does NOT recommend or verify the safety of this application) then someone has written an application to check several things including card compatibility. XNA Requirements checker.