I do not like Mountain Lion because some of my paid old software will not run. Is there a way I downgrade back to OS X 10.6 if I don't have the original install disks and I did not take a single Time Machine backup? I thought there was a restore partition that had the original OS. Thanks.
No, you can't.
Recovery partitions were first introduced in 10.7. Your only option is to acquire the OS X Snow Leopard installer "somewhere else", then completely erase your hard drive, and install 10.6 fresh.
It is advised to keep backups at all times, especially before doing a major system upgrade.