The curvature of a curve is an intrinsic property that measures the variation of the unity tangent direction. In general, the curvature varies along the curve. Two kind of curves have constant curvature: circles and stright lines, which have zero curvature.
The radius of curvature is the invers of the curvature and is the radius of the tangent circle at a point on the curve that fits better. Obviously, the circle and the curve have the same curvature at the point where they are tangent.

Check that the given formula is correct for planar curves.
Check that if a curve is the graph of a function y=f(x), then the points of curvature zero are the points where f''(x)=0.