Java Web Application AWT or Swing?

When developing a Java web-based application, is it necessary to use AWT?

I've been researching this question and the answers are confusing. Take for example, Runescape. The user visits the website and can play the game right there. They can also download a desktop application to play the game with.

If I am developing a game application, and my main window is created in a JFrame but all of my graphics are processed in a JPanel inside the main window, will I be able to implement this inside of a web page without much difficulty in the future? I want my game to be accessible simply by visiting the website and running the application.

Additionally, I've seen people reference "Java Web Start". Will this load the java application inside of the website, or does it install/run Java and execute the program (so if my main function opens a JFrame, it would appear on your taskbar). Perhaps I'm having some confusion between a Java application and a Java applet.