Question

what does center of curvature mean?

Answer 1

in geometry center of curvature of a curve is found at a point that is at a distance equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the center of curvature.

Answer 2

It is the center of the circle/sphere, of which the arc/part of sphere forms a part of.
An arc is a part of a circle.
You might be using lens/mirrors(both convex and concave)......their surfaces form the part of a sphere(like an arc in a circle)......And it is the center of that sphere(completed) which includes the surface of the mirror/lens is called center of curvature of the mirror/lens.
http://www.physicsclassroom.com/class/refln/u13l3a.cfm

Answer 3

In optics, the center of curvature is the point that would be the center of a sphere if you consider the mirror (concave / convex) to be part of the surface of a large sphere. The distance from the mirror to the center of curvature is called the radius of curvature and is equal to twice the focal distance (the distance from the mirror to the focal point). This also applies to lenses.

An example: http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Concave_mirror.svg/758px-Concave_mirror.svg.png