How do you control the security of Firefox browser in Yosemite to run educational Java app?
The issue isn't with Firefox or Apple. It is Java itself that is blocking the application.

The way to fix this is to add the website to the Exception Site List within Java Control Panel.


Open System Preferences and click on Java, which should launch Java Control Panel.
Select the Security tab and click the "Edit Site List..." button near the bottom.
Then on the Exception Site List window, click Add to add a new blank line and paste in the address for the website. Then click OK on the Exception Site List window and you should now have something like this...




If you close and reopen Firefox now it should prompt you to "Allow artist.info to run 'Java Applet'?" and if you click allow it should now load the applet.

If you've tried this and it hasn't worked, something might be remembering a previous decision. Try clicking the "Restore Security Prompts" button at the bottom of the Security tab in Java Control Panel, or clicking Firefox's plugin icon in the left corner of the address bar (the one that looks like a Lego brick) to get it to allow the Java plugin.

Adding the site to the exception list, closing and reopening Firefox worked for me with Firefox 37.0.2 on Yosemite 10.10.3 and Java 1.8.0_45.