Flashman, whether the code is ran or not depends by a number of factors. Typically a modern browser (like Firefox) should tell you, when you are on a website that there is an applet trying to run and ask you wheather you want it to actually run or not (you can normally answer something like "yes only this time", "yes, anytime I visit", "not this time", "never"). This preference is then stored in the user's profile and will determine the behaviour for the next visits.
This is one of the various reasons for which is important - security wise - to use different users even within the family (where surely trust should not be an issue).
Java applets - however - are relatively secure. The "logic" of an applet is the one that kestas calls "sandbox": the code is ran within a defined "virtual space" that is defined by java, not by the applet, so that the applet itself should be unaware and unable to access whatever exists outside of its little space. *SHOULD* because software is never bomb-proof, and occasionally somebody - normally a 14 years old geek with no social life ;P- finds out a way to exploit a minor mistake, and mess computers up. This is why some browsers ask you if you want the applet to run or not: because despite the best efforts to "sandbox", there is no 100% guarantee that will be safe.
HOWEVER: this was to explain the rationale of applets. The possibilities that somebody could use this to get your writings are fairly low, the possibility that one could use a general-purpose applet like the one running here to gain access to your machine, find your files, and get them is possibly close to zero. Surely I would be far more concerned about the possibility of somebody stealing my laptop.
If security is so much of a concern for you I would highly recommend to use a GNU/Linux platform and to use an encrypted home folder: GNU/Linux is by far the most secure OS around (because of its design) when it comes to networked computers and to keep users apart, the encrypted folder is a must in case somebody should gain physical access to your machine. If you are not much of a geek, you can contact your local linux group: LUG's normally organise regularly "installation parties", where they help people who want to try GNU/Linux out to install and configure it on their machines...
Hope this help.