There are only four possible neighbourhoods of three cells, and each one can be declared dead or alive. That is sixteen bits of choice, and there are two hundred and fifty-six rules you can build from them. Some of them do nothing. Some of them fall apart. A few of them make a pattern so complicated that people have argued about it for eighty years.
The rule is written as eight binary digits, one for each of the eight possible neighbourhoods, read from 111 down to 000. Rule 30 is 00011110, which means: a cell with a live right-hand neighbour becomes live, and a cell with a live left-hand neighbour becomes live unless its right-hand neighbour is also live. Everything else stays dead.
That single sentence produces the chaotic triangle on the left of this page. Nothing about it was designed to look like that; the pattern falls out of the rule the way a river falls out of a slope.
Stephen Wolfram classified the hundred and twenty-eight elementary rules into four families by what they settle into: rules that end in a fixed point, rules that end in a repeating cycle, rules that produce ever-growing chaotic regions, and rules that produce intricate non-repeating structure that may or may not be complicated.
Rule 30 is the famous member of the last family, and Wolfram himself conjectured that it is universal — that it can simulate any computation, given enough time and room. That has not been proved, and it is one of the great open questions of the subject, and it is being argued about in this very paragraph's neighbourhood.