First, the fun bit:
To use: enter a code into the box, and press the “Go!” button to see the pattern. Try out a few codes. Particularly, try 150, 110, 182, 22, 18, and whatever else you can think of. Also, make sure to try some of them with the “random pixels” setting, as well as the “1 pixel” settings. The “Complement” button finds the code which generates the same pattern, but with black and white reversed. You can enter hex codes into the box if you prefix them with ‘0x’, or octal if you use a leading 0. If that’s your cup of tea.
What am I looking at?
You’re looking at a time vs. space graph of a one-dimensional cellular automata system. Each row of the graph is a generation; black pixels represent live cells, and white ones represent dead cells. Each row is generated using a simple set of rules, taking into account the previous state of the cell and its immediate neighbors. The rule corresponding to the code is depicted by the row of t-shapes in the grey rectangle. Also, this is a circular graph - the left edge wraps around to the right, and vice versa.
What’s so cool about this?
It makes pretty pictures! On that front, check out rule 150 with random pixels as the starting pattern. If you make an n-pixel wide pattern, it’ll repeat every n/2 pixels if you have wraparound (which this does). Rule 30 can be used as a random number generator, and rule 110 can be used to build a computer. Yes, a computer - specifically a Turing machine, which is a machine which, given enough space and time, can be used to compute any computable problem. Pretty slow though, and a real pain to program.
How does the code map to the rule?
Well, since there are 8 rules which make up every graph, a chap named Wolfram came up with the idea of using the on-off pattern of the rules as an 8-bit binary number. For example look at the on-off pattern of rule 30: 
or 00011110. That binary number corresponds to decimal 30. The upshot of this is that each of the 256 rulesets can be described using a number from 0 through 255, which is quite convenient.
Enjoy!