How do I upgrade my OS X 10.5.8 software to 10.6.3?

Buy Snow Leopard install DVD from Apple

If you have a PPC Mac don't bother. It can't be upgrade to Snow Leopad.

Allan

Choose About this Mac from the Apple menu and check the processor.

If it's a PowerPC Mac, it's already running the newest OS it can.

If it's an Intel Mac, click here, install the DVD, and run Software Update.

Niel, you forgot the most important step.

Backing up your data before any install:

https://discussions.apple.com/docs/DOC-1992

You Can't. PowerPC Chips are no longer put in ANY Mac that is out there. 10.5.8 is the last OS that the PPC Chips will handle because it is written in PPC & Intel Code.

And if you choose to get an Intel Mac, don't forget to follow this tip for migrating data:

https://discussions.apple.com/docs/DOC-2295