Applet: Conway's Game of Life
Conway's Game of Life is a surprisingly simple yet illuminating illustration of the complex ways in which a system can develop, even under extremely simple rules. In this respect, it is comparable to various fractals - easy to code, but eternally fascinating.
This is one of the first Java applets I made, just for fun.
If you left-click inside the applet, the grid will be cleared and re-seeded with a new shape from the small library built into the applet. There is a 0.5% chance, at each iteration, that the applet will introduce a random shape somewhere on the grid. Right-click inside the applet to turn off the randomness - the foreground colour will turn red.
Note that due to the way applets tend to interact with the user, you may have to left-click once inside the applet to be able to right-click in it, thus ruining the grid. This applet needs a better interface.
If you're curious, take a look at the old and horrible source code.