Conway's Game of Life is a cellular automaton in which there are two states: alive and dead. A cell will become alive (be "born") if there are 3 alive cells surrounding it (its neighborhood), and die if anything other than 2 or 3 alive cells surround it. Diagonals are included as part of the neighborhood (this is known as a Moore neighborhood)
It's notable for generating long-lived, complex patterns from a random initial distribution. I've included some common modifications in the suggestion setup table below, like HighLife (B36/S23) and Seeds (B2/S). It's tricky to find interesting rulesets, as most configurations will end in a stable state within one or two steps.
Link to this automaton