This is a collection of data organised in a specific way, typically stored on a server that allows easy access, retrieval and updating of the data.
The DBMS is software that runs on the server to act as an interface and allow end users to communicate with the database and process the data. Note that the database as such is nothing more than a collection of data. However, when people talk about databases, they often mean the database and the corresponding DBMS that is used to access the database.
In most cases, database systems are the most efficient and practical way to store data, and that is why they are used by companies and organisations all over the world. If there were no database systems, we would not be able to store the masses of data we generate today. If data is the oil of the 21st century, then databases are something like the tanks in which this oil is stored.