If you have a motherboard with an nVidia chipset, can you use a graphics card from ATI? Can you use two of them in CrossFire mode? And if you use a motherboard with an ATI chipset, will there be a performance loss in either graphics or motherboard performance?

You can certainly use a single PCI Express ATI graphics card in an nVidia chipset motherboard. You can also install two of them, but only if you plan on driving two separate displays.
Motherboards with nVidia chipsets do not support ATI's CrossFire dual–graphics card technology.
If you're using an AMD Athlon 64 processor, then you'll need a motherboard with an ATI chipset, such as the Asus A8R-MVP or DFI RDX200 CF-DR.

ATI chipsets result in slightly lower CPU performance than nVidia chipset boards, but the difference is less than 4 percent, so it's statistically not significant. However, it's been documented that the ULi south bridge chip used in most ATI chipset boards does register slightly slower USB 2.0 throughput. Whether that's a problem or not is something only you can determine.

Note that if you have an Intel processor, motherboards using Intel's 975X chipset, such as the Intel D975XBX or Asus P5WD2-E motherboards, fully support ATI's CrossFire technology.