One row of cells and one rule decides the next row from each cell's three neighbours. Rule 30 churns out pure randomness, Rule 90 grows a Sierpinski triangle, and Rule 110 is powerful enough to compute anything. Try the presets.
Each of the 256 rules is an 8-bit number: bit abc says what the centre cell becomes when its left, centre and right neighbours are a,b,c. Rule 30 (bit pattern 00011110) generates randomness so good it ships in Mathematica's random-number generator. Rule 90 is XOR, producing the Sierpinski fractal. Rule 110 is provably Turing-complete. Rule 184 models traffic flow.