Help with java software? Not very good with software!?

Okay, I tried to register on a website, but it said I needed to have Java. I have a Mac OS X version 10.5.8. So I can update Java when I click on software updates. I clicked on software updates, but it's up to date already. The website is gosupermodel.com. Would I have to buy a different version of Mac to update java and play the game? I don't want to have to buy anything. I'll add detail if you need it.