The meaning of the curvature

Basically, the curvature of a curve at a given point measures how quickly the curve changes direction at that point. But what exactly does that mean? What "direction" changes.... is it an angle? Suppose the curvature is 2. How would I interpret that number as the rate of how a curve changes? There must be some physical meaning behind the number.

Another question - am I correct to assume that a vector function, when differentiated twice, can't denote direction in 3D as opposed as in 2D. In 2D, the acceleration denotes the rate of how the slope changes, negative or positive. But in 3D.... the acceleration function is basically denotes the rate of how the vector rate changes with no data based on 3D direction?

You getting my drift?

Thanks!