Chaos theory explains why some phenomena are unpredictable even though the phenomena are described by mathematical equations that are known perfectly well. For example, the weather seems impossible to predict accurately even though we have a very good mathematical theory that describes the weather. It was a great discovery (and shock!) to scientists in the 1970's when they realized that even very simple equations can generate extremely complicated behavior. That was the beginning of so-called "chaos theory". The main idea here is that chaotic systems are extremely sensitive to small disturbances of the system. These small disturbances (which are inevitably present in any real system) get magnified so much as to make predictions impossible.
Fractals are geometric objects that have a very complicated structure yet are remarkably easy to describe (and to draw with a computer). They are appealing to the eye because of their great amount of symmetry ( some fractals). Fractal-like objects were discovered in mathematics more than 100 years ago, but required the computer to bring them to life. Here the main idea is "self-similarity"; a fractal looks the same on all scales (if you look at a small piece of it and magnify it, it looks like the whole thing). Thus, a fractal is infinitely complicated. Nature is full of self-similarity: mountains, waves on the sea, craters on the moon,.... There's even "fractal music".
The connection between chaos and fractals are the strange attractors. To every dynamical system (i.e., every system or object that evolves in time) whether chaotic or not, there is a "phase space"; the collection of all possible solutions (or types of behavior) of the system. This is a collection of curves, say, in two or three dimensional space (think of the flow of water; each particle follows one of the curves). We can look at the geometry of these curves, that is, their shapes. You can imagine that if these curves have a complicated shape then the behavior of the corresponding solution will be complicated (smoothly flowing water vs turbulent flowing water). What could be more complicated than a fractal? It turns out that in the phase space of every chaotic system there is a strange attractor. It is an "attractor" because it attracts solutions (so solutions eventually become as complicated as the attractor), and it is "strange" because it has a fractal structure, and so is infinitely complicated. This is the cause of the "chaos" in a chaotic system. ( Some strange attractors.)
Finally, we will look at certain dynamical systems that arise when you iterate a simple function like f(z)= z*z + c. Here z and c are complex numbers. You can think of a complex number as a two dimensional number, so you can plot them on the two dimensional (x,y)-plane (that is, we write z = (x,y) where z = x + iy, and i is the square root of -1). Start with a complex number z. Then call z(2)=f(z), z(3)=f(z(2)), etc. This way we generate the sequence {z,z(2),z(3),...} of complex numbers by iterating the function f starting with the number z. We then ask the question: For which numbers z does this sequence go off to infinity, and for which numbers z does this sequence remain bounded? (The sequence goes off to infinity if the z(n)'s get arbitrarily large. However, if the z(n)'s remain less than some (constant) number for all n, then this sequence does not go off to infinity.) It turns out that for any number z, either the sequence goes off to infinity or it doesn't (this may sound obvious, but it isn't). Labeling numbers (the z's) black if the sequence starting with that number doesn't go off to infinity and white if the sequence does go off to infinity, gives us a black and white picture on the (x,y)-plane. Depending on the number c you use in the function f(z), these pictures can be exeedingly complicated. These are the Julia sets. Colouring the numbers black, red, orange, yellow, and white depending on "how fast" they run off to infinity, gives us a colour picture. These pictures can be very beautiful. (Some Julia sets.)
The Mandelbrot set is a way of ordering all the Julia sets (for all possible numbers c). The Julia sets are either one piece or are totally disconnected ("dust"). If we color black those c's whose Julia set is connected, and white those c's whose Julia set is dust, we obtain the famous Mandelbrot set ( the Mandelbrot set). Both Julia sets and the Mandelbrot sets have a fractal-like structure in the sense that they are infinitey complicated. Furthermore, if one looks closely at the Mandelbrot set one sees tiny replicas of Julia sets. There are many secrets of the Mandelbrot set that have yet to be revealed.