When you purchase a product, there isnt a FileMaker database per se. It is the application you are talking about. You would have to create a database, use one of the pre-existing templates, or use a pre-made file.
The FileMaker Pro Application is used to create the database. FileMaker Server is another application that hosts FileMaker database files ( create in a Pro version for multiple Pro clients to connect.
You will need at least one copy of Pro or Pro Advanced to create your database files first. Then you can use the server application to host those files.
I hope this helps clear some things up for you.