CLARIFICATION: The connection $\nabla$ used below is the Levi-Civita connection for the Riemannian metric on $M$ and *not* $N$.

First, assume that $N$ is a curve
and recall how to construct the Frenet frame of a curve.
Let $e_1 \in T_*N$ be a unit tangent vector along the
curve. If $\nabla_{e_1}e_1 \ne 0$, then, since $e_1\cdot\nabla_{e_1}e_1
= 0$, there isa a unique unit vector $e_2$ and function
$\kappa_1 > 0$ such that $e_2\cdot e_1 = 0$
and $\nabla_{e_1}e_1 = \kappa_{1} e_2$.
If, in turn, $\nabla_{e_1}e_2 \ne 0$, then, $e_2\cdot \nabla_{e_1}e_2 =
0$ and $e_1\cdot \nabla_{e_1}e_2 = -e_2\cdot\nabla_{e_1}e_1 =
-\kappa_{1}$. Therefore, there exists a unique unit vector $e_3$
and function $\kappa_2 > 0$ such
that $e_3\cdot e_1 = e_3\cdot e_2 = 0$ and
$$
\nabla_{e_1}e_2 = -\kappa_{1} e_1 + \kappa_{2} e_3.
$$
If, at each stage, $\nabla_{e_1}e_k \ne 0$, then this leads to a
uniquely defined orthonormal frame of vectors $e_1, \dots, e_n \in
T_*M$ along $N$, known as the Frenet frame and corresponding functions
$\kappa_1, \dots, \kappa_{n-1}$ such that
$$
\nabla_{e_1}e_k = -\kappa_{k-1}e_{k-1} + \kappa_{k}e_{k+1}.
$$
If $M$ is flat, then the vanishing of $\kappa_{j},
\dots, \kappa_{n-1}$ implies that $N$ lies in an affine subspace of
dimension $j$. The function $\kappa_1$ is usually called the curvature
of $N$, and the functions $\kappa_2, \dots, \kappa_{n-1}$ considered as
torsion functions.

This can be generalized to higher dimensional submanifolds
as described by Liviu. Here is a sketch of one way to do it:
If $n_1 = \dim N$, let $e_1, \dots, e_{n_1}$ be an
orthonormal frame of vectors tangent to $N$. Suppose at each point
in $N$, the vectors $e_1, \dots, e_{n_1},
\nabla_{e_j}e_i$, where $1 \le i, j \le n$, span an $n_2$-dimensional
subspace, where $n < n_2$. Extend the orthonormal frame to a basis $e_1, \dots,
e_{n_2}$ of this larger subspace. The generalization of $\kappa_1$ for a
curve to a higher dimensional submanifold is the second fundamental
form
$H_{ij} = H^\mu_{ij}e_\mu$, where $H^\mu_{ij} =
e_\mu\cdot\nabla_{e_i}e_j$ and $n+1 \le \mu \le n_2$. Now suppose that $e_i, \nabla_{e_j}e_i$,
$1 \le i,j \le n_2$ span an $n_3$-dimensional subspace of
$T_*M$. Again, extend the orthonormal frame to one that spans this
subspace.
Then one can define a torsion tensor $T_{\mu,\nu} = T_{\mu\nu}^\eta
e_\eta$, where $n+1 \le \mu,\nu \le n_2$ and $n_2+1\le \eta \le n_3$.
This process can be continued. If the resulting frame does not span $T_*M$, then
one might call $N$ degenerate. If $M$ is flat, this implies that $N$
lies in an affine subspace. Otherwise, one gets a higher dimensional
version of a Frenet frame. The frame is not unique, but the
the nested sequence of osculating subspaces obtained in this way are.