CONWAY'S GAME OF LIFE
An excellent starting point is here. To use the applet, click on the Enjoy Life button, then on the Open button. A menu of patterns will appear, click on one, then on the OK button. Read the description, then click on the OK button. Switch to the full screen mode, then click the Go button. Recommended as good illustrations of emergent patterns are GliderGunds and (especially) InfinityHotel. Also, click on `Patterns too big for the Applet link. This will let you access the page where a Turing machine constructed within the Game of Life is described.
Another nice set of applets is here and a site with much useful information is here (Click on the + symbols to access the various links). And, of course, the Wikipedia article is a good starting point.