I am trying to install the MAC OS X Snow Leopard upgrade, but it is not installing correctly. It seems to be perpetually stuck in a spin up cycle.

If you computer meets the system requirements
An Apple computer with an Intel Processor
1GB of Memory (RAM)
5GB of hard drive space (Storage)
DVD drive for installation

If your computer meet these requirements and is running 10.4 or later a direct update should work.
You can boot from the OS install DVD and under utilities run disk utility and repair permissions and the disk then try again

Still referring to my first post.
Remember Snow Leopard dose not support PPC processors.
Dose the computer meet the system requirements for the OS being installed?
If yes.
Boot into (AHT0 Apple hardware test here is how you need to use the disk originally supplied with the computer.
AHT will boot on many machines not able to boot an OS or OS Installer.