There are eight possible arrangements of three black or white cells in a row. For each arrangement, toggle the blue highlighted cell centered below it to choose whether it should be black or white. That choice is a rule, and all eight rules make a rule set that covers all possible combinations for a group of three consecutive cells. There are 256 (that's 28) possible rule sets. (By convention, the rule sets are numbered from 0 to 255).
The first row of green highlighted cells represents an initial state. These are the first cells that the rule set is applied to. Click on any of them to toggle their color between black and white. It's initially set to just a single black cell, but you can create any sequence of black and white cells. Each row of cells is then generated entirely by the row just above it, according to the rule set. So the first row generates the second row, the second row generates the third row, and so on. Everything is generated entirely from the initial state and the rule set. You can't directly toggle the color of cells in the lower rows.
Think of the row as a ring, with the left and right endpoints connected in a loop. Many rules create diagonal structures that wrap around the edges of the grid. Depending on the rule and the initial state, the patterns formed can be simple lines, elegant fractal geometries, or non-repeating chaotic structures. To explore the complex possibilities, view this page on a large desktop monitor, and try larger numbers (columns > 500, rows > 1000).