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 browser. How do I go about doing this?
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.