Milnor writes on p. 11
If $M'$ is a manifold which is contained in $M$, it has already been noted that $TM'_x$ is a subspace of $TM_x$ for $x \in M'$. The orthogonal complement of $TM'_x$ in $TM_x$ is then a vector subspace of dimension $m - m'$ called the space of normal vectors to $M'$ in $M$ at $x$.
But what does "orthogonal complement" mean here? $M$ is merely a manifold, not Riemannian.
In Milnor's book all manifolds are submanifolds of $\mathbf{R}^n$, so they inherit the Riemannian structure from $\mathbf{R}^n$.