How to explain fractals to a layperson and to someone with more math training? I have a Ph.D. in computational and theoretical chemistry with advanced but field-oriented knowledge of mathematics.
I am fascinated by fractals, but I am unable to understand them from the formal point of view. To my level of understanding, they look like a graphical rendering of an ill-conditioned iterative problem, where small variations of the initial condition lead to huge changes in the final result, but that's just what I got out of it with my current knowledge.
How would you explain fractals (such as the Mandelbrot set) to a layperson with basic mathematics knowledge from high school, and how would you instead explain it to someone which has more math training, but not formal. 
This question is collateral to a post on the Mandelbrot set I did on my blog some time ago. If you have any comments on what I was doing with my tinkering of the parameters (to get some keywords for further exploration), it's greatly appreciated. I would like to explain it better to my readers, but I am unable to do it. Thanks
 A: There are a lot of examples that are not hard to define rigorously, e.g. the Cantor set. This can be defined rigorously simply by taking a suitable intersection of nested intervals when the middle thirds are successively removed. It is easy to make all sorts of variants, constructing "Cantor-like" sets when you replace "middle third" with "middle fourth" or even with a construction where the ratio of the interval removed changes with the iteration. (This is important because you can then get a set of positive measure.) Leaving aside the remark about measure, all this uses nothing more than elementary mathematics.
One way to think of fractals is as of subsets of the phase space of certain dynamical systems. For instance, consider the Smale horseshoe. This is a two-dimensional map that can be visualized geometrically; this is explained in the Wikipedia article. Then the collection of points that stay in the square when you iterate this procedure both forwards and backwards is a fractal set, basically a two-dimensional Cantor set, which is very interesting from the perspective of dynamical systems (it's chaotic---one consequence of this is that knowing a point to arbitrary precision will not tell you what its orbit looks like later on with much precision).  So much of the interesting behavior of dynamical systems is on fractals. You can define a map of the interval into itself such that the points in the interval that stay in the interval no matter how many times you iterate the map is just the ordinary Cantor set, as well.
A: For a "high-level" explanation, I would say this: Fractals are surprisingly complex patterns that result from the repeated application of relatively simple operations/rules.
One of the easiest to visualize examples is the Koch snowflake, constructed by adding smaller triangles to each face of the figure at each iteration:

A more real-world example is the fern leaf. The DNA in a single plant cell encodes enough information to describe the structure of an entire leaf (and the entire plant, for that matter) without explicitly describing the location of each cell. Instead, the cells grow according to a set of simple rules that result in the self-similar appearance of the fern, even at smaller and smaller levels:

For a more complex mathematical explanation that still remains tied to the real world, have a look at the basic Ricker model of population growth and the resulting bifurcation diagrams:

(source: phaser.com) 
The x-axis on this graph is population growth rate and the y-axis is population density. Although it looks complex, All it takes is a handful of iterations of the basic formula on a hand calculator to see how the results can oscillate between seemingly random population levels.
A: That's easy.  I would buy them some of this stuff:

(Then I would tell them to stare at it for awhile.)  It's called Romanesco broccoli, and it grows like this for the same reasons, as explained in e.James' answer, that ferns do.
Alternately, I would tell them to check out Indra's Pearls by Mumford, Series, and Wright.
A: The link http://www.fractalsciencekit.com/types/classic.htm explains, in simple terms, the process of creating several different fractal types including Mandelbrot, Julia, Newton, and Orbit Traps.
A: The accepted answer restricts itself to fractals, which makes it correct but does not answer the OP's question as stated.
Fractals
The following are reflections on the task of explaining what a fractal is and isn't. (Here, M is to be considered the as yet undefined subject OP wants to explain.)


*

*If you consider M a set, leave out 0i. You are explaining a set, not a subset.

*M is a field, not a set.

*Understanding this difference helps understanding why M is not a fractal by Mandelbrot's definition. Anyone can visualize any iterative formula in any way they choose, it's completely ad hoc.

*This is where general confusion starts, because a Mandelbrot looks sort of like the other fractals. Quaternions look like fractals only because we choose to visualize slices, because the dimensions do not fit our visualization or geometry generation preferences. In other words, visualizations are not limited to projections etc, although they can yield interesting mathematical connections being less ad hoc.

*As with familiar geometries, to make a connection with physics, the connection must be proven, or it remains a mathematical abstraction. Such proof seems elusive, so I would tone down the connection to Nature in a presentation, even though reducing a coastline in dimension to a length creates a valid understanding of fractals, and growth patterns of plants match fractals well in the contexts of scale and subdivision (as presented in the accepted answer).
Teaching
Explaining a concept to someone for which it is advanced is a challenge. It means you must understand it all the better to not leave a step unexplained or use lax (incorrect) terminology which halts understanding at some point. It's very hard indeed, it's much easier to be this strict in writing than in a presentation, because there is time for reflection and feedback.
Most mathematicians are more familiar with arithmetic functions and successive approximation methods to find roots I think, and so may find trouble explaining this face to face if not equally prepared.
Understanding
From your qualifications, I think Weintraub's proof that M is a field will let you find distinctions on which to build an understanding of what fractals and Mandelbrots are.
I think the Coastline Paradox page is a solid foundation on which to build an explanation, and in the Iterated Function System section under the Fractals heading, you will find more fractals.
From these, understanding that only the boundary of the Mandelbrot field is a fractal might aid explanation.
