A fractal is an object or quantity that displays self-similarity on all scales. The object need not ehibit exactly the same structure at all scales, but the same type of structures must appear at all scales. An interesting, and paradoxical example is that of a coasline. The length of a coastline depends on the length of the ruler you measure it with. The smaller the ruler, the more detail of the shoreline is measured and the longer the coast.
The repetition of a form in smaller detail is known as an interation. Iterations can easily be seen in both fractal trees and fractal ferns. In a true fractal the iterations would continue to smaller and smaller detail if I had the skill and tools to produce it.
Many people are aware of fractals from the exploration of the famous Mandelbrot set. This is a set of equations that lead to a graphical display of infinite complexity. Computers have enabled the exploration of the boundry of this form that shows stunning complexity and beauty. Many programs can be found that allows the viewer to explore the set and the beauty of the colored graphics it can produce. Here is a link to one of these sites.