Database can be classified in three major categories.
Flat file database system. -- Old system very redundant, hence not much preferred.
DBMS -- A database that can store your data in rows and column and in ordered way but cant form relations between the tables. These are still in use but i won't suggest it.
RDBMS -- Relational database management system. A system that allows the user to maintain integrity of the data. It allows easy insertion, updation, deletion and backup of data.
There are a lot of database system in the world----- examples
DB2 by IBM
Postgres SQL -- Free and open source
MySQL --- Open Source and widely used in PHP sites
MS-SQL Server -- By Microsoft
MS-Access by Microsoft for general or small scale data handling
Derby --- By Sun Micosystem