It depends on how you want users to chat. Implementing a (1) 100% Web-based chatroom (no applet), or (2) users use applet from a Web page to chat? Or (3) users have dedicated clients (e.g. swing GUI) for chat?

I hope this is not college assignment. We don't allow posting these kind of questions here. But I will not post any code anyway as they are not something you can do with just a few lines of code and you should try to explore on your own.

(3) The GUI client can support any client-server socket protocol so this is the most traditional way to do this
(2) An applet can have limited networking capability so you can mostly use your own protocol as in case (3)
(1) Without applet, immediate refresh of message on Web page may be triggered by AJAX. You can use Tomcat (or even application server such as JBoss) to host an application server with state managed on it. However, to avoid excessive AJAX polling, you can use Jetty instead of Tomcat, that supports "continuations" so that it can push update to client instead of requiring polling from client side. Actually, the Jetty distribution has an example that exactly demonstrates how to create a chatroom using this method.