As with all such questions there is no general "best". If you are lucky, there might be a "best" for your particular application at the moment, though. For a full answer we will need to know a bit more about what you plan to do with your Linux database, and also how much you are willing to spend.
As a starting point, MySQL is free and very fast, but lacks some of the more complex database features like transactions, PostgresSQL is very popular, more powerful than MySQL and also free. Oracle is very powerful and includes lots of proprietary features (like
Java stored procedures!) but can be expensive.
Remember as well that there are
lots more databases available, a lot of which come with Red Hat, and many others are free to download.
