Yes, it is possible.
You have two choices. Do all the server code by you self, in which case, you'll have to defined the protocol and everything ( that is start from absolute 0 ) If that's what you want you have to start with Java sockets. That contains a very simple server to start with.
More advanced networking could be achieved using the Apache Mina project.
Servlets in the other hand, are some kind of "plugins" for an existing HTTP-WebServer usually referred as "servlet container". With servlets, you just provide a part of the functionality, and the existing servlet container, invokes it when it is turn.
The most usual servlet container is Apache - Tomcat and here's servlet tutorial
Finally as for distribution goes, if you go for the first approach, webstart is ok.
If you decide for the second, webapps are usually distributed via .war files.