Recall that the Mandelbrot is the set of points $c$ of the complex plane such that iterating the function $f(x) = x^2 + c$ starting from $x=0$ does not diverge to infinity.

The Julia set corresponding to a point $c$ of the Mandelbrot is defined as the set of points $z$ such that iterating the function f starting from $x=z$ does not explode to infinity.

Hence, by definition, to each point $c$ of the parameter space where the Mandelbrot sits, there corresponds a unique Julia set in the dynamical space.

The relationship between these two fractals is intimate and deep. In particular, points inside the Mandelbrot correspond to connected Julias, while points outside of it give rise to disconnected Julias.

Fractals are self-similar objects, that is, they appear the same at different scales. First theorized in the 17th century by the mathematician and philosopher Gottfried Leibniz, fractal shapes are common in the empirical world, with examples including plants, coastlines, human blood vessels and pulmonary vessels, DNA, and ocean waves.