Database: A database contains a collection of related items or facts arranged in a specific structure. The most obvious example of a non-computerized database is a telephone directory.
DBMS (Database Management System): A database management system is a collection of interrelated data and a set of programs to access those data. It allows people to store, access and process data or facts into useful information.
Actually to make large collection of data really useful, people and organizations use data management system which allows users to easily make use of the data.