You need to use a Facebook Account (Personal) to create an application. It can't be created from a Facebook Page. Everything in Facebook must be 'owned' by a personal account, so if you have a Facebook Page, you can create the application with the same user that created that Page.
Page's never own applications. However, a Page can show a tab in it's left menu that links to an application. Many Pages can link to the same application (since there is no ownership of it).
For example, we have a Facebook Page (
www.facebook.com/SourceCoast) and the Application that's used on this site. On the Page, in the left menu, is a link to "JFBConnect". That's our Application (just a page from our website) in a Page Tab. We have full instructions on how to setup a Page Tab below:
www.sourcecoast.com/jfbconnect/docs/facebook-page-canvas-setup
Hope that all helps, and makes sense. If you need more helps, let us know. It definitely can be confusing, we understand where you're coming from!
Alex