As you may know, OS X 10.8 introduces a new security feature called Gatekeeper which can conditionally disallow software downloaded from the internet from being launched on your machine. You can access the Gatekeeper settings in the Security pane of System Preferences.
Gatekeeper offers three "trusted source" security settings for launching downloaded apps:
Only launch apps from the Mac App Store
Only launch apps from the Mac App Store or digitally signed with an authorized Apple Developer ID
Launch anything downloaded from anywhere
The default Gatekeeper setting is Number 2. My guess is that the sort of people who visit this site will have lowered that to Number 3, "Allow from Anywhere".
[UPDATE: As of macOS 10.12, the third option is no longer available.]
Gatekeeper doesn't take notice of many of the types of AppleScripts available from this site so downloading and launching those scripts from here won't trigger Gatekeeper no matter what the security setting. AppleSctipt applications, which constitute the majority of recent applets and droplets, have been/will be codesigned with an Apple Developer ID.
However, it is still remotely possible that a handful of older scripts from this site may be quarantined by Gatekeeper and won't be permitted to launch after you download them. (This does not affect scripts that you have already downloaded and have launched at least once.)
What to do: Control-Click/Right-Click on the script in the Finder and choose "Open" which will present an option to override any Gatekeeper settings. Alternatively, you could lower your Gatekeeper security setting temporarily while you launch the script and then restore the setting afterwards.
Once you've successfully launched the script the first time, by whatever means, it will be ignored by Gatekeeper thereafter.
Update: additional new information posted here.