Java applets allow dynamic content to run within web pages. An applet runs on the client-side browser and is embedded in web pages. Applets have some advantages like less response time since they run on the client-side, and can run on multiple platforms. However, applets require a plugin to be installed on the browser. The lifecycle of an applet includes initialization, starting, painting, stopping, and destruction phases managed by the Java plugin. Graphics methods like drawString and drawOval are commonly used to display graphics in applets.