What does it mean for a curve to move in the normal direction? What does it mean for a curve to move in the normal direction?
Especially in the context of shape optimization and level set methods.
Does it mean that the boundary of a set is considered a curve and when this curve evolves through different shapes, then each point in the boundary is moving either to its inward normal or outward normal direction.

It means that you have to describe this "movement" through a family of implicitly defined functions under the form:
$$\tag{1}f_t(x,y)=0$$
where $t$ is a parameter that you can consider as a time, and that the gradient 
$$(\partial f/\partial x,\partial f/\partial y)$$
indicates the direction of evolution of the curve ; the coefficient $k(x,y)$ of the normalized gradient you will attribute will give the local evolution speed (the intensity) of theses curves. Rather often, this speed is governed by a certain power of the local curvature, and the curves appear as solutions of differential equations or partial differential equations.
When you examine with another point of view, you can consider the family of curves defined by (1) as the level sets of a certain surface. This explains that "level sets" is a good keyword for the subject. 
