The Java source for this applet can be found here. This was written in 2000 to be a little fun doohickey on an "About" box.
The basic idea is that the dots are particles on the surface of a bubble. The dots repel each other. Acceleration and damping are simulated. The particles are confined to the surface. The movement of the group reminds me a bit of "flocking" (for more on flocking simulations see Boids). Notice that if you let the applet run long enough, often a "current" will appear... the particles moving together in the same direction.
This applet requires Swing components and Java 2 (v1.3). If it doesn't work in your browser, there are a couple of options. The only one that worked for me was to use appletviewer. To do this, download and install the Java 2 SDK and run appletviewer on this URL.
An easier way, if it works, is probably to get the Java Plug-in, which you may already have, and visit the Java Plug-in version of this page.
If you want to do more work, and fix your browser so it can run all the latest Swing apps, download the Java 2 SDK and try setting your CLASSPATH variable to point to the right places. See Make Your Browser Swing! Note: I couldn't get either of the last two methods to work on my machine. (D-:)
Here's the applet again in a smaller size and with some different parameters: