Library or Flat-File Database:
Think of a library: a huge pile of books (data) just sitting their waiting to be used. Although the books may be organized (sorted), there is no relation between the books. The database contains the information as well as an index. The index serves the role of the libraries card catalogue, hence the name "flat file." (Think of an index card).
Relational Database (RDBMS):
In a relational database, several different types of data are stured in several different files (or tables), but the data items are linked through common fields or attributes. For example, Drexel maintains a database of students, but it also maintains a database of Professors. Although these two groups of data are usually kept separate, the can be related by classes taken/taught or through majors/departments.
Hierarchical/Network Database:
In a hierachical database, the data is organized in levels. For example, the files on your computer are arranged in a hierarchy. This type of organization is characterized by the "many to one" relationship. A more advanced version of this relation is the "many to many" relationship of the network database. These types of databases are difficult to visualize (and maintain)!