Hi,
As it is already explained very well by given answers.
Database is nothing but collection of data, information like files, tables(cells, records) etc.
Database management system as its lateral meaning shows its a system to manage database.
it includes following functionality:
->how to store data
->how to display data
->how to manipulate data

for example SQL Server data base management system
Database : SQL database : collection of tables
System Fucntionality:
DDL-> how to store data: query like create, alter, insert...
DML-> how to manipulate data: query like update, select...
DCL-> how to control data : query like Commit, rollback, savepoints...

Hope you would be getting some idea.

Regards,
Neel