How it works
One dimension first. A row of cells, and a rule that reads three of them — a cell and its two neighbours — to decide what that cell becomes. Three cells give eight possible neighbourhoods, and a rule is a choice of output for each, which is eight bits: 256 rules in total, and Wolfram’s numbering is nothing more than those bits read as a byte. Each new generation is drawn as the next row down, so the sheet is a picture of time.
The whole interest of the thing is that this is enough. Rule 30 makes an unpredictable stream out of a single black cell and was used as a random number generator for years. Rule 90 draws the Sierpiński triangle exactly. Rule 110 is Turing complete — it can, in principle, compute anything. Most of the other 253 rules are stripes or nothing, which is itself the lesson.
Two dimensions is the same idea with eight neighbours instead of two, and a rule that is two sets of counts: how many live neighbours bring a dead cell to life (B), and how many let a live one stay (S). Conway’s Life is B3/S23. Moving one digit gives Maze, or Coral, or Seeds, none of which look remotely like Life. Because a two-dimensional run has no spare axis to draw time on, this one is drawn after all the generations have finished — and the "when it was alive" colouring is the way to get the history back, since it shades every cell that ever lived by the last generation it managed it.