Let $M$ be an $n$-dimensional Riemannian manifold and $\Sigma\subset M$ an embedded submanifold of dimension two, i.e., a surface. Its second-fundamental form ${\text{II}}:\Gamma(T\Sigma)\times \Gamma(T\Sigma)\to \Gamma(N\Sigma)$ takes values in sections of the normal bundle. If $\{N_i\}\subset \Gamma(N\Sigma)$ is an orthonormal basis of sections of the normal bundle, we can decompose the second fundamental form as
$$\text{II}(X,Y) = \sum_{i=1}^{n-2} \text{II}^i(X,Y) N_i,$$
in which $\text{II}^i:\Gamma(T\Sigma)\times\Gamma(T\Sigma)\to C^\infty(\Sigma)$. We can then associate a $2\times 2$ matrix of components for each $\text{II}^i$ and therefore we have a collection of $(n-2)$ such matrices.
In $n=3$ there is just a single matrix. Its eigenvalues are defined as the principal curvatures $\kappa_1$ and $\kappa_2$, and from them we can construct the Gaussian curvature $G=\kappa_1\kappa_2$ and the mean curvature $H = \frac{\kappa_1+\kappa_2}{2}$.
What happens in this more general case? What is the appropriate generalization of the principal, Gaussian and mean curvatures to $n>3$? What is the geometric data contained in the second fundamental form analogous to these quantities $\kappa_1, \kappa_2, G, H$ in $n=3$?
The obvious thing would be to consider for each of the $\text{II}^i$ an associated pair of $\kappa_1^i$ and $\kappa_2^i$ with its associated $G^i$ and $H^i$, but this seems quite random and doesn't seem to be the right quantities to consider.
First, let's review what happens when $n=3$. In that case, there is a unique (up to sign) normal vector $N$ (also known as the Gauss map), and at each point $p \in \Sigma$, the second fundamental form is a symmetric bilinear tensor $$ \operatorname{II}: T_p\times T_p \rightarrow \mathbb{R}, $$ where for any $v, w \in T_p\Sigma$, $$ \operatorname{II}(v,w) = g_M(v,\nabla_wN) = -g_M(\nabla_vw,N), $$ where $\nabla$ denotes the Levi-Civita connection of $g_M$ (and not $g_\Sigma$). If $(e_1,e_2)$ is an orthonormal basis of $T_pM$, then the mean curvature is defined to be $$ H(p) = \operatorname{II}(e_1,e_1) + \operatorname{II}(e_2,e_2) $$ and the Gauss curvature is $$ K(p) = \operatorname{II}(e_1,e_1)\operatorname{II}(e_2,e_2) - (\operatorname{II}(e_1,e_2))^2. $$
When $n \ge 3$, then at each $p \in \Sigma$, the second fundamental form is the bilinear map $$\operatorname{II}: T_p\times T_p \rightarrow N_p, $$ where $N_p\subset T_pM$ is the subspace of vectors normal to $T_pM$ given by $$\operatorname{II}(v,w) = \pi^\perp(\nabla_v w), $$ where $\pi^\perp: T_pM \rightarrow N_p$ is orthogonal projection. Then the formulas above, with scalar multiplication replaced by the inner product defined by $g_M$, still hold: \begin{align*} H(p) &=\operatorname{II}(e_1,e_1) + \operatorname{II}(e_2,e_2)\\ K(p) &= g_M(\operatorname{II}(e_1,e_1),\operatorname{II}(e_2,e_2)) - g_M(\operatorname{II}(e_1,e_2),\operatorname{II}(e_1,e_2)). \end{align*} $K$ is still the Gauss curvature and a scalar function intrinsic to $\Sigma$. However, $H$ is no longer a scalar function. It is a section of the normal bundle, i.e., $H(p) \in N_p$. It also is still what appears in the variational formula for the area of $\Sigma$.
ADDENDUM: If $n > 3$, the principal curvatures are also no longer well defined as scalars. However, for each unit normal $\nu \in N_p$, you can define the principal curvatures for that unit direction to be the eigenvalues of $\nu\cdot\operatorname{II}$.