When I hear someone mention any particular RDBMS, they generally refer to it as a database. What is intended by the term "management system"?
A database is simply someplace to store data -- A file that you dump names & addresses into is technically a database.
A database management system includes all the other tools that make it easier to work with (manage) your data -- a query language, a way of defining structure, methods of indexing the data for faster access, etc.
Generally speaking you don't have a database without a management system these days.