Click on the Apple logo in the menu bar>About this Mac and it should tell you what OSX you are running.
If it is not 10.6.8 you will need to use the retail install disk to upgrade it. If you do nnot havr this disk (the one with the picture of the cat on it) you would need to call apple and order one or check here http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard
If you have the disk try doing a back up first and reinstalling, power off the mac and slide the disk into the slot, then power up holding the C key and press in the the disk, continue holding the C key till you see the progress bar then let it boot and follow the prompts for install.
If it appears that 10.6.8 is installed you may try this following update direct from Apple's site and see if it installs, if not try the reinstall mentioned above and then run this update, (these are stand alone updates with the installer package, so I would set Safari to download them to the desktop and install them from there. Safari preferences>general, save downloaded files: Desktop) then run software update to see what other updates that
you will need. Write down the name and version of the recommended updates and go to this link and type in the update name and version to find it (them) and do the same.
Hope this helps
If you do not have this install disk, this is what you need to update from 10.5 to 10.6.
You can also try booting up holding the OPTION key and see if has a Snow Leopard boot option (maybe a disk partition not in startuo disk) Also disconnect all external, non-essencial devices
Or do the reinstall if you have the correct disk.