A database (which we will call BDD for convenience) is basically a collection of information organized to be easily searchable, manageable, and up-to-date. In a database, data is organized in rows, columns, and tables. They are indexed to easily search for information. Whenever new information is added, the data is updated, and eventually deleted.
Currently, the market is dominated by DB2, SQL Server, Oracle, and IBM. On Windows, SQL Server is generally the BDD of choice, while Oracle and DB2 are the most popular on Mainframe / Unix or Linux ecosystems.
Some of the most popular:
Additional Resources: