Julia's idea was to observe the behavior of a complex number (represented as a point in the plane) under iteration of a function; that is, a value of the function is fed back into the formula for the same function.
As an example, consider the function f(z)=z^2 - 1/2. Certain starting values of z generate a sequence of complex numbers which remains bounded. For instance, the starting value z = 0 produces the sequence:
f(0) = -.5 f(-.5) = -.75 f(-.75) = .0625 f(.0625) = -.4961 f(-.4961) = -.25
and so on. One coloring scheme would be to color all such points with one fixed color, say black.
Other starting values of z give rise to a sequence which escapes to infinity. For example,
f(2) = 3.5 f(3.5) = 11.75 f(11.75) = 137.56 f(137.56) = 11922.94
and so on. These z-values are then colored with various colors and shades according to how fast the sequence approaches infinity. The Julia set is the boundary between these two sets of points.