Does Java SE (Standard Edition) offer a way to make its programs work online beside the Applets? Or is the applet the only way to do that?
You could use Webstart to allow users to launch your Java application from a website and have the application communicate back with a server. As with applets, Webstart applications run in a sandbox by default and hence you should look into signing your application jar to allow it to communicate over the network.
You could also consider GWT or Servlets / JSP if your aim is to write a web application rather than deploy a standalone app. It all depends on what you're trying to achieve.
Java is trying to make a comeback on the client-side with a new technology called JavaFX.
Not sure if that is not just an applet under the hood, though.
And then there is WebStart, which launches Java applications from a website (although they then run outside of the browser).