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.
IBM provides good set of information and tutorials about building RESTful web service with Java (Link). After getting your web service running, you can deploy it to Amazon. Take a look at AWS Elastic Beanstalk.