HyperSQL is the world's most popular pure Java database. It is also known as "HSQLDB". It is integrated with tons of Java applications, including Open Office.
If you are running Linux, the best database management systems are MySQL, Apache Derby, PostgreSQL, HSQLDB (HyperSQL), or Ingres. However, there are many types of database management softwares, so please be more specific with your question if this was not the answer you were looking for.