Who database engine is better, in performance and usability?
In case of webmasters choosing shared hosting, the database available to you from hosting service is dependent on their choice. Most likely MySQL and PostgreSQL are common database that you can find in shared hosting services. When you move to VPS and Dedicated server hosts, they offer you database of your choice for both managed and unmanaged hosting. They offer pre-installed database servers based on their abaility to resolve technical issues. So some may prefer mysql over any other database or postgresql over any other. It is purely personal choice for hosts and unless you go with the route of your own VPS and Dedicated server, there is not much of choice for you to choose database.
In case of programmers choosing the database, it purely depends on his ability to work with respective db tools and error solving capability. MySQL and PostgreSQL database have much better support and you can easily find solutions to the issues by just googling the problems. Many other databases which are not used extensively in server environment have less documentation, blog posts available and you're likely to hit walls during problem solving. Again, this all depends how efficient programmer is while handling database.
If you're not a programmer or webmaster and just thinking about database during installation of web app like wordpress or joomla then consider using the one available in your cpanel of your hosting. Issues of performance and stability of database will be taken care of by your hosts.You don't have to think much about it.
My rough observation : MySQL and PostgreSQL - these are the ones often preferred by linux based hosting services. SQL Server (MS) is dominant on windows hosting. SQLite is rarely used for webapps and usually selected by those who know how to get it working over webserver. Oracle is heavily used for enterprise apps. MongoDB is used by ruby on rails programmers for their rails applications. Cassandra and CouchDB rarely noticed with web apps or if any programmer using it.