From your description, It might be a web app. Unlike apps from the App Store, web apps are just links to sites that are designed to look like apps. The get rid of the browser interface, and store enough local data to make them a bit more usable than a normal web site.
If it is a web app, it probably prompted you to use the Add To Home Screen action. This will add an icon to the home screen that looks just like an app icon; however it is just a shortcut to Safari. But with a web app, you won't see the Safari interface.
At any rate, you won't have gone through the App Store, and that means that you won't see that app in your purchased list.
If you did go through the App Store (hard to miss), then it's a different issue. My best guess is that the app is actually an iPhone app. iPads will run most iPhone apps, though they won't look quite right.
In that case to back to the Purchased tab in the App Store. Near the tip right it will say iPad Apps. Tap on this and change it to iPhone Apps. You should then see any iPhone apps you've purchased from the iPad store.
There are four kinds of apps in the App Store.
1) iPhone apps that will only run on an iPhone, usually because the app needs some feature that only exists on the iPhone.
2) iPhone apps that will work on an iPad, but are not optimized for it. The screen with be the same as for an iPhone, either blown up larger, or shrunk to a portion of the iPad's screen.
3) Universal apps that are optimized for both iPhones and iPads. They switch interfaces depending on what device they are on.
4) iPad apps. They only work on an iPad.
Note: Apps for the iPod Touch would be either type 2 or 3.