If you make the dots stand for smaller sub-divisions of the graph you can get more a detailed pattern. In most cases after a certain point there is no more detail to see. A fractal graphic is one where you can increase the level of detail for ever, and it will always show you more.
The Mandelbrot set is a particular group of numbers, z squared + c, where 'c' is a complex number (one which includes the imaginary number i). If you iterate the equation (get the result, use the result to repeat the equation, use the result again... and so on) for some numbers the result will shoot off to infinity, and for other numbers the result will level out. The numbers which level off are called the Mandelbrot set.
A graph of the Mandelbrot set looks like this (black = a number in the set, white = a number not in the set). You can colour in the white section, by using different colours to show how quickly the equation shoots off to infinity. Then it looks like this.
The graph of the Mandelbrot set is fractal. You can zoom in for ever, and see more detail. Computer power has made it easier and easier to display that process.
This is the best Mandelbrot zoom I have ever seen. In virtual-reality terms it is the size of the known universe, and takes three and a half minutes.
ETA - sorry I have not replied to comments, work has suddenly become very pressured again