I am relatively new to Java and have a Java application consisting of a couple of packages and a number of classes. I want to be able to run this application in a web browse
I've written some kind of JVM which precompiles Java classes to native JavaScript. Here is an HelloWorld example which runs java programm in the browser:
https://github.com/neo-expert/jsjvm_helloworld
it has also a WebGL demo which renders a 3D cube.
Java Web Start is a good technology for deploying Java applications over the web. You can start the application from a web page link, although the application runs outside of the web browser frame: Java Web Start Tutorial.
Java applets allow you to embed a Java application in a web page. It has some serious drawbacks for larger applications.
Servlets (and Java Server Pages) are appropriate technologies for server-side generation of web pages (and other web content) but these technologies won't help you to deploy an existing Swing-based Java application, unless you are prepared to replace the user interface.
If you want a really fancy web-browser based application written in java, then you should learn gwt.
See the section of the Java Tutorial on how to build and deploy applets.
The easiest way for you will be to use a servlet. What you need: