I am trying to learn Socket programming in Java to make a game I created playable via network. I have never done much network programming so the subject is a bit confusing. In php I use apache to test my aplications on a local server. What do I need to install to be able to run network applications with Java?
You don't need to install anything special. Socket has been in the standard JDK since version 1.0.
See socket tutorial to get started.
If you want to test how the communication works between processes, then simply start >1 processes in the same machine. In principle it works similarly over any network.