How to upgrade my browser on 10.6.8
It really depends on which browser you are using...  I would recommend not using Safari on 10.6.8, since it is unsupported by Apple at this point, and the older versions of Safari have known compatibility / security issues.

The best way to "upgrade" would just be to install a third party browser (Chrome or FireFox, which I linked to below) and those issues should go away.

To clarify, Safari is a fine browser on later versions of OS X; specifically ones that Apple still releases security patches for.



Safari

On Snow Leopard, Safari cannot be upgraded past version 5.1.10. Wikipedia article

FireFox

It appears that you can install the latest version of FireFox, according to the system requirements for FireFox 37 page on mozilla.org

Google Chrome

Google Chrome's official download page says that the latest version runs on 10.6.x Google Chrome Download Page