A "Julia Set" is a set of complex numbers, given a complex number C as a seed. Each complex number Z0 is plugged into the formula
Zn+1 = Zn2 + C
and iterated indefinitely. If the results remain bounded, then that Z0 is in the Julia Set for that C.
When plotted on the argand plane, Julia Sets' appearances vary in complexity from simple circles and ellipses to fantastic psychedelic patterns. Since it's been proven that the formula will not remain bounded if, for a given n, arg Zn > 2, computer programmers like to color each number that isn't in the set based on the lowest n for that Z0.
Julia sets are the brainchild of Gaston Julia, who plotted several of them (by hand!) while recovering from a wound he received in World War I. Some wound. Julia knew nothing of "chaos" or "fractals"; he was probably trying to prove the Riemann Hypothesis. But without the infinite patience required to crunch numbers for several weeks, these "mathematical monsters" remained largely unexplored until computers came along to do the calculations and make pretty pictures.
You may have heard of the Mandelbrot Set, which uses the same formula, but varies C for Z0 = 0. The Mandelbrot Set (which, when plotted on the Argand plane, looks like a Mathematics professor who has eaten too many Twinkies) can be used as a map of the Julia sets1.
Some Julia sets are connected, that is, around each point in the set you can draw a closed curve, within which all of the points are in that Julia set; the rest are totally disconnected, which means around each point you can draw a closed curve, within which the point is the only member of the set. If a complex number C is in the Mandelbrot Set, the Julia set generated from it is connected. If C is not in the Mandelbrot set, its Julia Set is disconnected.
As it turns out, only the boundary has a fractal character; each point in the interior of a connected (filled) Julia Set converges to the origin. Because of this (and adding to the confusion), some mathematicians use the term "Julia Set" to mean the frontier of a Julia Set using the definition above, preferring the term "Filled Julia Set" for those.
Curiously, each Julia set strangely resembles the part of the Mandelbrot Set its seed point came from.
1Some mathematicians define the "Mandelbrot Set" as a subset of C x C, all points (Z0, C) for which the formula given above remains bounded. If we plot this "Big Mandelbrot Set" in a 4-dimensional analogue of the Argand plane, each plane Y = C (where C is some complex number) contains an analogue of the Julia Set for C. The point (x, C) is in the "Big Mandelbrot Set" if x is in the Julia set for C. Each Julia set's plane is "parallel" to all of the other Julia Sets' planes. The plane X = 0 (which is orthogonal to all of the Julia planes) contains an analogue of our "map", the more familiar Mandelbrot Set. (0, y) is in the "Big Mandelbrot Set" if y is in the "Little Mandelbrot Set".