Database for Java?

Well, instead of a PHP script, you can communicate to the database with... Java DBC. As your server is already in Java, just implement the database access in Java instead of adding the additional overhead of calling a PHP api and so, it'll be easier and faster. Also, there are some libraries and frameworks to make your life easier (I'd recommend ActiveJDBC, but you can find some more through Google). Also, you can use MySQL or whatever database you prefer, so no problem with that.

I don't understand what do you mean by "LibGDX environment", but I don't think LibGDX has any integrated solution to access databases, but probably someone developed a third party solution, so it's worth to check.

Well, currently my server runs in a LibGDX framework/environment. I will just use the exe/jar from the desktop project to run the server so I do not expect to run into problems. But since there is no need to run the server from a android device I could probably ditch LibGDX and just use Java with Kryonet and add a DB extension to the mix to keep the server project clean.

For security reasons, you don't want to allow your clients to access the database directly.

That is why i do all the database communication through the server side.

Okay, I misunderstood this:

"Is it possible to setup somekind of DB and sent/receive queries directly to/from the database."

To connect to a database from Java all you need is a JDBC driver, establish a connection, and you can issue SQL queries right away.

Pretty much all big databases out there (MySQL, PostgreSQL, Oracle Database, etc) have a JDBC driver.

You can follow this tutorial on how to use JDBC http://docs.oracle.com/javase/tutorial/jdbc/

Okay, I misunderstood this:

"Is it possible to setup somekind of DB and sent/receive queries directly to/from the database."

Yeah, what i meant was without weaving in a php script through a http request. So directly, but from the server side.

If you want a lightweight SQL solution then JDBC is the way to go.

If you like JDBC but don't like how tedious it is to use then look at the Spring framework's JDBCTemplate module, which fits exactly that purpose.

If you want to move away from SQL and towards ORM (Object-relational mapping) then looks into JPA and/or Hibernate.

As for databases themselves, any will do, including MySQL if that's what you're comfortable with.

Maybe Cassandra would be a good choice in this context?

It seems that there will be much more writes than reads, and that's the kind of use case Cassandra was designed for.
Datastax driver for Java is pretty straight-forward, but there are also drivers for other languages: http://planetcassandra.org/client-drivers-tools/

If your usage of the database is quite low and your server can do its own locking (i.e an exclusive lock of the whole database whilst saving some player stats), then I might recommend SQLite.

There are a few java bindings knocking around like: http://code.google.com/p/sqlite4java

SQLite is cool because it is very easy to migrate between servers and does not rely on sockets so your game server can access it directly (obviously this means you want the SQLite database on the same server).