At its most essential level the DBMS architecture can be divide into two parts: the back end and the front end.
The back end is responsible for managing the physical database and giving the essential support and mappings for the conceptual, internal and external levels. Other advantages of a DBMS, such as integrity, security and access control, are also the duty of the back end.
The front end is really just any application that runs on top of the DBMS. These may be applications given by the DBMS vendor, the user, or a third party. The user deals with the front end, and may not even be aware that the back end exists. This interaction is complete through Applications and Utilities which are the major interface to the DBMS for most users.