You need to do one of the two: keep it updated or get rid of it. Either is fine for most users. Yes, it is safe to update Java. See my article should I update Java? for complete answers to these questions.
.EXEfiles, these "applets" can not access or delete your personal files unless they ask for and are given express permission to do so. In the real world, users hardly ever give permission for this, so applets generally don't ask.
As of this writing, Java is usually (though not always) included as standard equipment on Windows PCs. If you choose to use Java applets on your site, you can invite your users to download the Java plug-in from Sun's website, using the "Get It Now" button on that site.
Java ExamplesJava can be used on PCs for both applets (interactive features inside web pages) and stand-alone applications (non-web programs like Notepad or Excel— these are not written in Java, they are just examples of applications).
Java applets have been almost completely replaced by Flash, but there are occasional jobs for which Flash is ill-suited. An example is my own Fracster mandelbrot set explorer, which lets the user explore an interesting mathematical function in a graphical way. While not impossible in Flash, this sort of mathematically intense, pixel-by-pixel display is better done in Java.
There are also many older applet games, such as Atari's official Asteroids applet, that simply haven't been rewritten in Flash.
The Azureus file-sharing application is a good example of a popular stand-alone application written in Java.
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