One of my favorite examples of cellular automata is Conway's game of life. 4 simple rules on a grid create complex systems (oscillators, glider guns, even a general purpose computer). It's a zero player game–meaning that the game play is determined solely by its initial state.
1. A live cell dies if it has fewer than two live neighbors.
2. A live cell with two or three live neighbors lives on to the next generation.
3. A live cell with more than three live neighbors dies.
4. A dead cell will be brought back to live if it has exactly three live neighbors.
Here is a quick simulation I made in javascript:You can manually click some of the boxes to create the starting conditions, or you can just click randomize.