The rule of a CA is the collection of which nbhd configurations give rise to a live cell in the next generation. For example, suppose the rule is this: if a cell is dead and both its left and right neighbors are alive, then the cell becomes alive in the next generation; all other configurations give a dead cell in the next generation. Here is the only configuration that gives a live cell in the next generation.
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To deduce the next generation from the current, look at each cell in the universe and compare its nbhd to the rule. If the nbhd matches the rule, the cell is alive in the next generation; if the nbhd does not match the rule, the cell is dead in the next generation.
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