Password Alert tries to accomplish its goals in two ways. First, Password Alert will show you a warning message if you type your Google Account password into a non-Google site. Second, you will be encouraged by Password Alert to change your password if you are repeatedly using your Google Account password on non-Google sites.
According to Google's Online Security blog Password Alert will make Chrome save “scrambled” version of your Google Account password. And according to the same post, it only remembers this information for security purposes and doesn’t share it with anyone.
Applications for Education
Google Apps for Education administrators can push Password Alert onto users. Administrators can also set up alerts to be notified when users use their passwords on non-Google Apps sites. Administrators are also able to force users to change their passwords if they use their passwords on non-Google sites. Administrators can learn more about this on the Google Apps Updates blog.
Folks who don't use Chrome can take steps to secure their Google Accounts by using two-step verification, frequently changing their passwords, and avoiding using the same password for multiple accounts.