A defining characteristic of a fractal is self-similarity. This means that the fractal looks similar at different scales. For example, a fractal may be a large triangle made up of smaller triangles, which are made up of even smaller triangles. A fern leaf appears to be made up of smaller fern leaves each of which is composed of even smaller fern leaves. While mathematical fractals can be exactly self-similar, fractals in nature are only approximately self-similar.
The dimensions of geometric constructs are whole numbers. A line is 1-D, a circle is 2-D and a cube is 3-D. Fractal shapes have fractional dimensions. To understand this, consider what happens when the dimensions of a geometric figure are doubled. This results in 2^d copies of the original figure where d is the dimension of the figure. When the length of a line segment is doubled, then two copies of the line segment are produced.
Fractals look complicated but you can use very basic rules to generate them. These rules are often iterative, that is repeated. For example, you can generate the Koch snowflake pattern by starting with an equilateral triangle and drawing a triangle with sides that are one-third of the original on the middle third of each side of the large triangle. You then draw in smaller triangles in the middle third of each side of the resulting figure. This rule is applied repeatedly to each new figure generated.
Fern leaves and river networks are not the only fractals in nature. Jagged coastlines are also fractal shapes. As you zoom in on the image of a coastline, the indentations look similar at different scales. A chambered nautilus is a particularly beautiful example of a shape composed of a pattern repeated at different scales. Cauliflowers, stalagmites, leaf veins, lightning, mountains and clouds are all complex self-similar fractal shapes.
Technology is now catching up with nature and using fractal shapes. Fractal antennas can receive broadband signals in a limited space. Optical fibers bundled in fractal patterns create waveguides with very low distortion. The fractal nature of Internet traffic sets network design parameters.