Are you comparing Intel integrated graphics cards with ATi (not AVI) and nVidia?
If you are then nVidia by far; they're in the lead with the 8800 series, and two of my desktops both have nVidias that perform quite well (FX5200 and 8600GT)
I have an ATI and an nVidia. So far, I can't tell the difference. They are both far better than any Intel integrated graphics though. I wouldn't touch those with a 10-foot pole.
I changed the AVI to ATI. I don't think you can compare the integrated graphics with external graphic accelerators. nVidia wins hands down on this battle. The 8800GTS is definitely better than any one by the ATI Series.
AVI is a video format not a graghics card, i hate both intel and nVidia but i like ATI because it was bought by AMD and is quite good, i have ati radeon xperss 200 series