I purchased a copy of UFO: Aftershock in this weekend's sale, downloaded all three installer parts, and then installed it. When I tried to run it, I got the attached error. I'm running Windows 7 (64-bit) on a laptop with an Intel graphics display and an NVIDIA graphics card; I've tried disabling UAC, reinstalling UFO:AS, and running it in compatibility mode for Windows XP. So far, nothing has worked. What should I do?

UFO:AS has no problems with Windows 7 (as long as you install in a non-UAC controlled folder), but it does have a lot of problems with Intel graphic cards. The error you've attached is basically the game trying to run on the Intel card, and failing.
Check your computer's documentation on how to force an application to use the NVIDIA card, and only the NVIDIA card, or disable the Intel card altogether.
Unfortunately, I'm not sure how to do that, so I can't help any further.

Is your NVIDIA chipset new enough to support DirectX 9? Check this list to make sure:

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