A collection of data organized in a certain way, typically stored on a server, that allows for easy access, retrieval and updating of data. The DBMS is a piece of software that runs on the server to act as an interface and enables end-users to communicate with and manipulate the database. Note that the database as such is nothing but a collection of data. However, often when people speak of databases, what they are referring to is the database and the corresponding DBMS used to access it.
In most cases, database systems are the most efficient and practical way to store data, and so they are used by companies and organizations around the globe. If there were no database systems, we would not be able to store the masses of data we are generating today. If data is the oil of the 21st century, then databases are something like the tanks used to store that oil.