You are absolutely correct! The Reminders app does not exist in your system, at least under that name. It is one of the "new" features of Mountain Lion. If you want the Reminders app, you will want to upgrade your system to Mountain Lion (OS X 10.8.x)
You can do that very easily through your App Store. It's only 19.99 for that upgrade
Then, the Reminders app will be automatically installed with the rest of the upgrade to your system software.