My fav database is PostgreSQL. Both MySql and PostgreSQL are totally free.
> Another two "more professional" databases: Firebird
> and SapDB.
And by "more professional", you mean????
From Java, MySQL is probably the best-supported database out there. The driver is more mature than most databases (open source or otherwise), and in fact version 3.0.0 of the driver passes all of the JDBC compliance test other than stored procedures (which MySQL doesn't have yet).
Of course, I'm biased (as I'm the JDBC driver developer), but I've used PostgreSQL, Oracle and SQLServer from Java and other technologies, and MySQL is definitely the easiest of them to use.
As far as paying for MySQL, you only need to pay for it when you distribute it with software that is not GPL-compatible.
Since hosting a website is not considered distribution, you have to pay nothing to use it.