The curvature $\kappa$ of a curve $C$ at a point $P$ is defined as the reciprocal of the radius of curvature $\rho$ of $C$ at $P$. Conversely, the normal curvature vector of a curve $C$ lying on a surface $S$ at $P$, denoted by $\mathbf{k}_{n}$, is defined as the vector projection of the curvature vector $\mathbf{k}$ of the curve $C$ at $P$ onto the normal $\mathbf{N}$ of the surface $S$ at $P$; i.e, $$ \mathbf{k}_{n}=(\mathbf{k} \cdot \mathbf{N}) \mathbf{N} $$
And its magnitude, $\kappa_n=\mathbf{k} \cdot \mathbf{N}$, is the normal curvature of $C$ at $P$.
I wonder what is the need to define this new type of curvature, $\kappa_n$, and what its geometrical meaning is. Does it have an intuitive meaning similar to that of the curvature $\kappa$, which can be understood as the reciprocal of the radius of the osculating circle to the curve at point $P$?
No, the normal curvature is a characteristic of the surface and not of the particular curve. Any two curves with the same tangent direction at $P$ will have the same normal curvature at $P$, but will generally have wildly different curvatures.
You can recover the normal curvature in direction $v$ as (up to sign) the curvature of a particular curve — namely, the normal slice of the surface in direction $v$ (i.e., the intersection of the surface with the plane through $P$ spanned by the normal and $v$).
See section 2.2 of my differential geometry text for more details. Be warned, though, that I use $\mathbf N$ for the principal normal of a curve and $\mathbf n$ for the surface normal.