Database stores more than just strings and integers, it can store multimedia data as well like pictures and videos. Database management system is the software platform that is used to manage the database while the database is simply the data items that are organized in a specific way. The combination of database management system and the database is called database system.
The database model (database schema) is what breaks down the specific details on how the database will look. There are three types of database model that breaks down the design in different levels which are conceptual, logical, and external data models. Conceptual is big-picture view that helps business users perceive data. Conceptual is a “implementation independent” model and consists of ERR (enhanced entity relationship) model or object oriented model. Logical data model is a “implementation” specific model that helps map conceptual data model toward specific environment and is database management specific. External data model focuses on the needs of groups or users.
Catalog is what stores the definition of these different data models and synchronizes it and data definition and manipulation languages are what gets used (example: SQL) to express it.