Self-similarity, which is a consequence of a power law relationship, and non-integral dimension are the main characteristics of fractals.
A fractal (from the Latin fractus, derived from the verb frangere, to break, to divide infinitely) is an object or a geometrical construct (or set of such objects) whose parts are somewhat identical or statistically similar with the whole through transformations that include translations, rotations and zooming. A structure is fractal if its finer shapes are indistinguishable or statistically similar to coarser ones, and thence there is no dominating scale.
The Koch curve is such and object, and in order to identify it as fractal it is customary to determine its fractal dimension. That the fractal dimension of the Koch curve is greater than unity, which is the Euclidean dimension of a line, should not be surprising; it simply indicates the high degree of roughness of the Koch curve with respect to a line.
The Koch curve tends to to fill out the two-dimesional plane, and thus it is reasonable that its dimension should be more than one, although certainly less than 2. Winding coastlines or the winding path of river channels can be simulated with a Koch curve, since typical coastlines have fractal dimensions around 1.25.
Consider the unit line segment but use the following rule: divide the line segment into three equal segments and discrad the middle one. Iterate. The result is the so-called Cantor's dust (named after German mathematician Georg Cantor). This is a fractal object that consists of just a line of disjoint points, hence the name dust. The fractal dimension, i.e. the ration logN/log(1/R) is a constant, equal to 0.63093. Intuitively it is reasonable to expect that since the rule simply pokes holes into a line, the resulting fractal dimension should be less than unity, as in fact it is.
Other fractal structures of similar nature can be obtained by poking holes in a plane or in a three dimensional volume. For instance, to construct the so-called Menger sponge (named after Austrian mathematician Karl Menger) we start with a solid cube of unit side. The rule is: scale down the side of the cube by; 1/3, to obtain 27 cubes of side 1/3, discard the mid-face cubes and the one at the center. Iterate. The fractal dimension of the Menger sponge is 2.72684..., reasonable less than the original dimension of the cube because the sponge has voids at all scales.