Conway's Game of Life
Wikipedia has quite a bit of info on Conway's Game of Life. With a very simple set of logical rules, very complex universes can be created. Test out your universe by setting your own rules for life, birth and death. The original Game of Life uses a 23/3 rule (2 or 3 neighbors stays alive, 3 neighbors creates new life, all others die) There is quite a bit of information on Cellular Automaton on Wikipedia, as well.