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The eight boxes below represent a rule. The rule governs how to color any pixel in a row, based on the state of the pixels in the row above. The three smaller boxes within each larger box represent the states of pixels on the previous row: the pixels to the left, directly over, and to the right of the current pixel on the new row. If you check the checkbox, the rule will color the pixel on the new row, if the three pixels on the previous row are in the states shown.
This section from Wolfram's book explains it all better than I can.