A database is an organized collection of data. It is a structure which can hold raw facts and files and can be structured into records and tables which have attributes with structural and value-based constraints.
It could be as simple as a physical notebook where a user writes down necessary data and maintains records, or it could be a tabulated flat file, or an Excel file.
A database management system is essentially software.
This software is an amalgamation of the actual database along with a management suite wrapped around it to offer various advanced tools and services.
These tools and services help the user to efficiently create, store, manipulate, manage, maintain, display, and safeguard the data in the database, to suit the business needs and processes.
A DBMS could be a simple desktop one like Microsoft Access, a full-fledged network-supported sophisticated commercial relational database like Oracle, a Web 2.0 No-SQL database like MongoDB, an open-source platform-independent one like SQLite, or even be an object-oriented one (OODBMS).