What does the Matrix digital rain effect represent?

The green Matrix "digital rain" that infested millions of PC screensavers in the late 90's was originally supposed to be a simply design "bookend", meant to reflect the heavy rain of the final battle in The Matrix Revolutions. The audience was supposed to see the rain not as failing water, but as the code of the Matrix itself falling from the sky as the Matrix was destroying itself under the weight of Smith.

That was the idea anyway, but ideas don't always translate cleanly into images. Rain is difficult to photograph (it often just disappears) so the filmmakers were forced to use such heavy rain drops that the set was completed drowned out by the noise, and the actors reported that they could barely see. I still count it as a success, the Matrix code is one of the most iconic and well-recognized designs from the film.

What the Matrix Code Represents

This is nothing like code in a modern computer. You'll notice in the code itself that the individual letters constantly change, and code overlaps itself in endless layers. It is a computer trying to represent the code of life, and the possibility that the smallest piece of the universe contains the sum total of all information. The opening of The Matrix Revolutions explains the complexity quite well:

An explosion of yellow light, which we know represents the Spirit World. Is this the birth of AI, or possibly God, or just the "Big Bang" of the Matrix Universe? I suspect all 3 at once.

Fractals form as the yellow light undergoes "cosmic expansion." Fractals like these (and most of the fractals we are used to seeing) come from an ultra-simple mathematical equation, the Mandelbrot set.