Discrete Quantized Cellular Automata
The page and most of the related pages concern cellular automata, specifically those that have discete "cells" each of which has one of a finite number of possible values (and thus is "quantized"). My use of discrete and quantized in the title emphasizes the fact that these are a subset of a larger set of systems that simulate continuous systems, and/or cells that can have an infinite number of possible values, and possibly both. For example, the Gray-Scott reaction-diffusion system is typically simulated on a finite grid of discrete cells, with values quantized to the precision of some fixed-size floating-point or fixed-point number. Even though that is technically a discrete, quantized system, it is treated as a continuous real-valued system and the patterns produced are usually indistinguishable from a "perfect" continuous real-valued simulation.
Many readers will be familiar with Conway's Game of Life. This is a cellular automoton with two values per cell ("on" and "off", or "1" and "0", or "alive" and "dead"). Each cell looks at its eight neighbors (a Moore neighborhood) as well as its own state to determine what it will do next.
Another important example is the von Neumann cellular automaton, which has 29 (or sometimes a few more) possible states per cell and is capable of producing complex machines that encode their own structure and reproduce themselves.
Types of Transition Rules
Totalistic rules have their own page.
Different Ways to Arrange the Cells
Hexagonal grids have their own page.