Below is one of the simpler cellular Automatas called Langton's ant. Each tick, if the ant is on a white cell it turns 90 degrees clockwise and if it is on a black cell it turns 90 degrees counter clockwise. It then swaps the colour of the cell it is on and takes a step forward.
Somewhat amazingly it turns out that Langton's ant is something called a universal turing machine. This means that any computation you can do on a computer can be done by Langton's ant by giving it the right initial setup of black and white cells and recording its output after some length of time.