The Curve deformation allows you to deform parts of a character using a Bezier curve. It is different from the Bone deformation in that the lines connecting the points are curvilinear. Hence, Curve deformations are typically used to deform parts of a character that are flexible, such as hair, mouths, eyebrows, or torsos. They can also be useful to animate limbs in a rubber hose style, where arms and legs are animated like extensible, flexible tubes rather than articulated bone structures.
By manipulating the curve's points and their Bezier handles, you can change the curve's direction, shape and length. A drawing rigged with a curve deformation does not merely bend, it will deform along its entire length to match the curve deformation's shape.