Mac shuts down when trying to download software update
Since your Mac doesn't start up anymore, try if you can do an upgrade install from an installation disk. See http://support.apple.com/kb/PH6606. It should keep user files and settings in place, but it often resolves issues like this. If you don't have backups, you can use Disk Utility to save a disk image of the drive to an external drive. See http://support.apple.com/kb/ht1553.

10.6 doesn't have a recovery partition, so you will need to use the older tools for booting and fixing your startup problem.