What do I need to create a RESTful API Server in Java?

You can use Spring controller for building a restful server. You can run it on tomcat or jetty doesn't matter.

Check this url : http://static.springsource.org/spring/docs/3.0.0.M3/spring-framework-reference/html/ch18s02.html

Tomcat and Jersey are easy to get up and running. I've had some issues with Tomcat 7 and Jersey, but with Tomcat 6 it was straight forward.

This tutorial is quite easy to follow. It's a bit old, but the principle remains the same.