An explanation of fractal dimension.
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One technical note: Its possible to have fractals with an integer dimension. The example to have in mind is some *very* rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!).

The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose «Hausdorff dimension» is greater than its «topological dimension». Hausdorff dimension is similar to the box-counting one I showed in this video, in some sense counting using balls instead of boxes, and it coincides with box-counting dimension in many cases. But its more general, at the cost of being a bit harder to describe.

Topological dimension is something thats always an integer, wherein (loosely speaking) curve-ish things are 1-dimensional, surface-ish things are two-dimensional, etc. For example, a Koch Curve has topological dimension 1, and Hausdorff dimension 1.262. A rough surface might have topological dimension 2, but fractal dimension 2.3. And if a curve with topological dimension 1 has a Hausdorff dimension that *happens* to be exactly 2, or 3, or 4, etc., it would be considered a fractal, even though its fractal dimension is an integer.

See Mandelbrots book «The Fractal Geometry of Nature» for the full details and more examples.

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