This is a chapter for the book I wrote for my Computational Modeling class at Olin, taught by Allen Downey.
I describe the famous two dimensional cellular automata known as the Game of Life, first discussed by John Conway. I outline the basic rules of the "game", describe various simulation techniques (including the HashLife algorithm), and discuss different pattern classifications (still-lifes, guns, gliders) and examples. Finally, I talk about what it means that the Game of Life is "Turing-complete", how to use the game as a computer, and how one could construct simple logic gates (AND, OR, NOT) in the game.
The logic gate implementations were first described by Jean-Phillipe Renard.
Rennard, Jean-Phillipe. "Implementation of logical functions in the game of life." Available online: http://arxiv.org/ftp/cs/papers/0406/0406009.pdf