Right now I have OS X 10.5.8. I downloaded 10.6.7, but when I try to install, I receive an error “System does not support.”
Can I download 10.6.6 and install it instead?
Those will not work on 10.5, You are at the highest version of 10.5. You will have to buy a copy of OS X 10.6. There is no illegal way to go from 10.5 to 10.6, doing it cheaply is another matter.
If you want to buy a copy, this would be the best place to start: