A database is a storing system that collects organized data, to make some works easier like searching, structure, and extend.
In web development, most databases use the relational database management system (RDBMS) to organize data and programming in SQL. Some databases, however, don't follow the former mechanism to organized data, which called NoSQL.
Some famous server-side RDBMS are MySQL(or MariaDB which is a fork of it), SQL Server, and Oracle Database. On the other hand, some famous NoSQL examples are MongoDB, Cassandra, and Redis.
Browsers also have their own database system called IndexedDB.