From Mavericks to Snow Lion: how to get back?
There is no "Snow Lion"—Snow Leopard, Lion, and Mountain Lion strike me as possibilities—but regardless, the answer is the same: you can roll back with your Time Machine to an earlier version of OS X:


Enter Recovery by rebooting and holding command+R.
Select "Restore from a Time Machine backup".
Select a backup from before your Mavericks upgrade.


Unfortunately, you may not be able to use all of today's data from a pre-Mavericks installation.  Some Mavericks applications (such as Mail.app) will upgrade their databases.  That data will not be able to be understood by pre-Mavericks versions of those applications.  If you haven't had any significant data changes since your upgrade, I would recommend a wholesale reversion back to your preferred OS.  If you have, perhaps you could roll back anyway and copy newer files from your Time Machine individually, provided they are not in a format your old OS can understand.

Apple TS article: OS X Mavericks: Revert to a previous OS X version