A fractal is often defined as a geometric shape that is self-similar, that is, whose magnified parts look like a smaller copy of the whole. The term "fractal" was coined by Benoit Mandelbolt in 1975 from the latin term fractus meaning "fragmented" or "irregular".
The basic concept
This concept can be explained in a commonly used analogy involving the coastline of an island.
The perimeter of the island would grow as you decrease the size of your measuring device and increase the accuracy of your measurements. Also, the island would more or less self-similar (in terms of becoming more and more jagged and complex) as you continued to shorten your measuring device.
Wikipedia, Fractals Page
Cynthia Lanius, Cynthia Lanius' Lessons: A Fractal Lesson
CoolMath.com, Math of Fractals