What is a "database"?
I will quote Dictionary.com, as I take this as the meaning of database:


  a comprehensive collection of related data  organized for convenient access, generally in a computer. 


Under this definition, you can consider a database anything from a full-fledged RDBMS (SQL Server, Oracle, etc.) to a basic flat file.  If it stores data, it technically can be considered a database.

Now, like most things in our modern world, there's the accepted meaning of a name.  And in the case of database, that will vary from person to person.  A lot of people think of a database solely as an entity managed by a data system.

It is worth noting @FrustratedWithFormsDesigner's comment:


  card-catalogues would also count if you removed the "...generally in a computer".


I agree with that statement, and I don't necessarily think that a database needs to live in a "computer" or any electronic device.  A card-catalogue is a perfect example of a non-computerized database.