First, a distinction should be made: a fractal is one thing, and certain methods for constructing particular fractals are another.
Loosely, a fractal can be described as an object which is self similar at different scales, that is, "zooming in" repeatedly leads to the same curve. An interesting property of fractals which is sometimes used to define them, is that one can assign a non integer dimension to them. For instance, a smooth curve has dimension 1, but a Koch snowflake is in a certain sense closer to being a two dimensional object, and we can assign it a non integer dimension of $\sim1.26$. Intuitively, A Sierpinski carpet is even closer to a 2D object, and indeed we assign it a higher fractal dimension, of $\sim1.89$.
As to Mandelbrot's famous set, the idea is as follows: To check whether or not a (complex valued) point $c$ is in the set, start with $z_0=0$, and iterate. When the series stays bound, $c$ is in the set. When the series diverges, $c$ is not in the set. (try $c=-1,0,1$ for yourself, and see what you get). For instance, is the point $i$, i.e. $(0,1)$ in the Mandelbrot set?
$$z_1 = 0^2 + i = i,\quad z_2 = i^2 + i = i - 1,\quad z_3 = (i-1)^2+i=
-i$$
$$z_4 = (-i)^2+i = i-1$$
Thus, the point $i$ leads to a bound repeated loop, and is therefore in the Mandelbrot set (i.e. the black area in most drawings).
The best really introductory textbook on FOL (first-order logic) that I have ever seen is "Language, Proof and Logic". It is really designed for beginners; it teaches not only the syntax and semantics of FOL (not as mathematical objects but truly at a beginner's level), but also teaches a complete Fitch-style deductive system for FOL. Beware of using any 'introductory texts' without checking thoroughly, as many of them that I have come across have fundamental conceptual flaws! I did not check 100% of LPL, but I didn't notice any significant issue so far.
After your target audience has learnt everything in LPL, they can easily move on to the other texts mentioned in this thread. Hannes' text, for example, starts by giving some simple structures from ordinary mathematics and explaining how we use FOL to axiomatize them.
Best Answer
I highly recommend the book that Maisam referenced in the comments below your question: Mendelbrot's The Fractal Geometry of Nature. It is widely acclaimed, and likely available at a library, public and/or academic.
You might be interested in some of the links in this "given the pattern, find the fractal" math.SE post: In particular, you might want to check out the link to The Algorithmic Beauty of Plants, which is a "book" whose chapters are available on-site for downloading in pdf form.
Of particular interest might be the chapter entitled: Modeling of Trees, and Fractal Properties of Plants.
For a vast array of examples of fractals, and a categorization of some major types, see the commercial website Fractal Science Kit (free for the browsing!)
Also, a link of interest might be the Yale Website for a class in Fractals. This page on "forgeries" might be of particular interest. E.g.: