This is often called the dual cone. Wikipedia has this illustration (here $x=0$ and $C$ is neither convex nor contains $x$, but the concept is the same):
Think of the dual cone $C^*$ as the set of directions such that starting from $x$, you don't move away from any point of $C$ (initially). When $C$ is itself a cone, $C^*$ consists of the linear functionals that are bounded below on $C$. This suggests that the concept may be relevant in linear programming and it is, as the article Conic optimization indicates.
Dual/normal cone is the same thing in the world of convex cones as polar set is in the world of convex sets.