Before we get into any more detail, we need to cover some basic terminology that will help you understand the unique qualities that fractals posess.
All fractals show a degree of what's called self-similarity. This means that as you look closer and closer into the details of a fractal, you can see a replica of the whole. A fern is a classic example. Look at the entire frond. See the branches coming out from the main stem? Each of those branches looks similar to the entire frond. They are self-similar to the original, just at a smaller scale.
These self-similar patterns are the result of a simple equation, or mathematical statement. Fractals are created by repeating this equation through a feedback loop in a process called iteration, where the results of one iteration form the input value for the next. For example, if you look at the interior of a nautilus shell, you'll see that each chamber of the shell is basically a carbon copy of the preceding chamber, just smaller as you trace them from the exterior to the interior.
Fractals are also recursive, regardless of scale. Ever go into a store's dressing room and find yourself surrounded by mirrors? For better or worse, you're looking at an infinitely recursive image of yourself.
Finally, a note about geometry. Most of us grew up being taught that length, width and height are the three dimensions, and that's that. Fractal geometry throws this concept a curve by creating irregular shapes in fractal dimension; the fractal dimension of a shape is a way of measuring that shape's complexity.
Now take all of that, and we can plainly see that a pure fractal is a geometric shape that is self-similar through infinite iterations in a recursive pattern and through infinite detail. Simple, right? Don't worry, we'll go over all the pieces soon enough.