Applications downloaded via the iTunes Store are meant for iOS devices, not your Mac (they wouldn't show up in either Launchpad or in /Applications/. You can find the location of your iTunes library by opening iTunes and going to Preferences > Advanced. Inside that folder, you should see a Mobile Applications folder, where all apps are stored.
If you did mean the Mac App Store, it's possible the application download was interrupted. If you go to the Purchases tab, all of the applications you have purchased would be listed, and you should be able to download and install any that are not already marked as such.