There are lots of open source database available. Here is the list of most popular open source database.

1) MySQL

http://www.mysql.com/

2) PostgreSQL

http://www.postgresql.org/

3) Firebird

http://www.firebirdsql.org/

4) GNU SQL Server

http://panda.ispras.ru/~gsql/

5) ScimoreDB

http://www.scimore.com/

6) DXstore database system

http://www.dss.bc.ca/dx/

7) SAP MaxDB

http://maxdb.sap.com/

8) HSQLDB

http://hsqldb.org/

9) CloudDB

http://www.clouddb.org/CloudDB11/

i hope this are useful for selecting the Open source database.