I think he means whether to get a Radeon or a GeForce graphics card, and you're right, you need a Radeon or GeForce for Dolphin really, as other stuff like integrated graphics do not have an adequate feature set to emulate the GC's graphics.
Well, first you have to find out what you computer can use. If you have an AGP slot, then get an AGP card; if you have a 16x PCI Express slot, then get a PCI Express card (not likely); or if you don't have either then it's PCI only.
Once you have this sorted, then you need to concentrate of card features. Dolphin makes use of shaders, so I'd strongly recommend a Direct X 9.0 compliant card. This means a Radeon 9500 or better, or GeForceFX 5200 or better.
As for ATi Vs. GeForce, it really doesn't matter which you go for for Dolphin as of right now. Both types of cards will display the graphics correctly, as they are fairly evenly matched.