One Dimensional Cellular Automata
A one dimensional cellular automaton is a discrete mathematical model for simulating complex systems. The environment is a linear strip of cells where each cell can have one of several possible states. The simplest interesting example, is a binary state cellular automaton, for example the states might be interpreted as alive and dead. The state of each cell in the next stage, or time step, is determined by its current state and the states of the cells on either side of it.
How the configuration of the three cells (self and neighbors) impact a given cell depends on the selected rule. A rule is a mapping from current cell configurations to a single cell state. For example, a rule might specify that a cell is alive in the next stage if it and its neighbors are all currently alive and is dead otherwise.
Elementary Cellular Automata
An elementary cellular automaton is a binary state (e.g. 0 and 1), one dimenstional cellular automaton. With two possible states for each cell and three "parent" cells, there are 23 possible parent configurations. Here they are:
111110101100011010001000
Each of these configurations can lead to a live cell or dead cell. Thus there are 223 = 256 rules for elementary cellular automata. Each rule is basically just a number that, when written in 8-bit binary form, determines how the parent configurations map to states for each cells. For example, rule 30 (or 00011110), looks like this:
111110101100011010001000
00011110
Where the top row is the parent configuration and the bottom row is the state of the cell with that parent configuration in the next stage.
See this WolframMathWorld article for more.