Thanks for pointing out to me that I neglected to update my profile when I updated my OS. I was running 10.5.8 up until a few weeks ago, when I upgraded through Snow Leopard and into Mountain Lion on the iMac in question, which is now running Version 10.8.2. Obviously that's very relevant to a question like this and I thank you for answering as if my profile were correct but pointing that out.
To others and general readers in this thread, what happened was I had no Java installed upon updating to Mountain Lion. To be honest, I'm quite happy that Apple would have disabled this problematic Java program to protect a user like myself who may or may not hear the latest news, but doesn't know exactly what to do about it. In fact, on another site I posted defense of Apple for doing this automatically to protect its community of users.
Unfortunately, I was told that a new program I would be working with for a job required Java to be installed. It wasn't in my System Preferences pane and I had to visit the Java site and download the program. Then I had to go into Java's own preferences pane (via the System Preferences pane) to configure it to my work's tech support guy's instructions. (I did ask at the time about the Java problems and the earlier Homeland Security warning, and he wasn't remotely concerned, but then, this was before yesterday's news. And of course it's my own personal computer, not office property!)
After writing my question, I actually got a software update from Java themselves, apparently, and downloaded that security fix from them. Not long after, I got a notice that the Apple Software Update had updates for me. It said Java and iTunes and I clicked on the update before I realized that I probably already had the Java update, but I just let it do its work. Fortunately, I didn't get an alert reporting that they had removed malware, so I suppose I didn't have any to begin with, but better safe than sorry.
For the record, I will be disabling Java as baltwo suggested, once this project is over; I did fine without it before installing it recently, and don't need one extra thing to worry about or rush to my computer to install security fixes for left and right.
Thanks to all for their responses!