Mac security problems
A feature of OS X Mountain Lion is the ability to run only signed applications - that is, applications from Apple's app store - out of the box.  This is a configurable setting, however, and can be changed in the settings screen you seem to have already uncovered.  If you set the "Allow applications downloaded from" option to "Anywhere", your Mac should run applications downloaded from the internet as well.



Do be aware of the security implications of this, and consider changing it only when necessary (and reverting the setting after you've installed the downloaded application which you trust).

For more information, and screen shots, see:
http://www.maclife.com/article/howtos/how_tweak_settings_gatekeeper_mountain_lion