So I suppose fractal dimension is a misleading term, it's more fractal ratio. Or an indicator how fractal an object is.
While fractal dimension does indicate how fractal an object is, it's also, in a sense, an extension of the standard concept of dimensions.
This video gives a pretty good explanation:
https://www.youtube.com/v/g-csmdpq39A&rel=1&fs=1&hd=1I'll also try my hand at explaining it. Here goes...
Take a straight line (a 1D object) embedded in a 2D space. If that line has length x , it can be covered completely by taking x circles of diameter 1, and placing them end-to-end along the length of the line. Now if you multiply the line's length by k, it will take k times as many spheres to cover it (specifically, x*k spheres).
Now try embedding the 1D line in 3D space. Similarly, a line of length x can be enclosed by x
spheres of diameter 1 (such that each point on the line is either inside a sphere or on the edge of a sphere). Multiply the line's length by k, and it takes k times as many spheres to cover it. Same rule as in dimension 2.
In general, for any non-fractal 1D-object, such as a line, embedded in an n-dimensional space, if you multiply the object's length by k, it will take k times as many n-spheres to cover/enclose it.
Now let's embed a finite plane (a 2D object) in a 3-dimensional space. Let's say this plane can be completely enclosed by n spheres of a fixed radius. If you multiply the plane's size by k, it will take k
squared spheres to enclose it, as the area of a plane has a quadratic relationship to it size.
In general, for any non-fractal 2d-object, such as a plane, embedded in an n-dimensional space, if you multiply the objects size by k, it will take k^2 times as many n-spheres to enclose it.
Now, if you consider fractal objects, the rules change a bit. Let's imagine a space-filling curve, such as the Hilbert curve, filling a finite plane. The curve itself is 1-dimensional, but when iterated to it's limit, it's essentially equivalent to a plane. The number of n-spheres (where n is the dimension of the space it's embedded in) needed to cover scales quadratically with the absolute size of the curve. SO a space-filling curve, although 1-dimensional object, has fractal dimension 2.
A similar thing can be said about all fractals. Most fractals are not space-filling like the Hilbert curve, yet due to their roughness, they "fill" more "space" than a non-fractal object of the same dimension. Take the Koch Snowflake. It's dimension is 1.26186. That means that, if you scale it up by factor k, the number of n-spheres required to cover it will scale by k^1.26186. So it's somewhere inbetween a line and a plane.
This sphere covering approach is essentially the same thing as box-counting.