We live in the 3-dimensional world of width, length, and depth. You probably know that a plane (length and width) is 2-dimensional, a line (length) is 1-dimensional, and a point is 0-dimensional. We can visualize an object in each of those dimensions. However, can you visualize a 1.2618-dimensional object? Probably not.

Think about fractional dimension this way. As a line-segment gets more bends and kinks in it, it becomes less of a 1 dimensional object, and grows towards becoming a plane.

However, we're going to be accurate and not say this bendy kinky curve is 2-dimensional. It's only (say, in the case of the Koch Curve) 1.2618. That makes sense because it is more than 1-dimensional, but not quite 2-dimensional.

- Abstracted from http://library.thinkquest.org/12740/netscape/discover/page6.html

Benoit Mandelbrot made the term Fractal to indicate objects with Fractional Dimensions.