What is Mandelbrot set and why it is shaped like that, what does it mean?

Answer 1

Take a value, square it again & again... if you number was >1, it goes off to infinity. If it's less than 1, it goes to zero. If you plot this on a line, you get a DOT at x= 1.00000....
Do this again on an x-y graph, and you get a perfect Circle of radius 1.0000....

Now start again. This time, Square a number AND add a small (constant) offset. Try -1 for a start.
Suddenly, instead of a simple circle, you get an odd shape. The Mandelbrot.