A curve $C$ on a surface $S $ is a set of points $\set{ \b{x} \in S : t \in (a, b)}$, continuously parameterized by $t \in (a, b) $; that is, $\b{x} : (a, b) \to S$ is a continuous function.
The curve $C = \set{ \b{x}(t) : t \in (a, b)}$ is differentiable if
$$\dot{ \b{x}} (t) = \der{ \b{x}}{ t } (t) = {\Vce{ \dot{ x_1 }(t)}{ \vdots }{ \dot{ x_n }(t)}}$$
exists for all $t \in (a, b) $.
The curve $C = \set{ \b{x}(t): t \in (a, b)}$ is twice differentiable if
$$\ddot{ \b{x}}(t) = \pderw{ \b{x}}{ t }(t) = {\Vce{ \ddot{ x_1 }(t)}{ \vdots }{ \ddot{ x_n }(t)}}$$
exists for all $t \in (a, b) $.
The tangent space at a point $\b{x}^* $ on the constraint surface $ S $ is the set
$$T(\b{x}^*) = \set{ \b{y} : D \b{h}(\b{x}^*)\b{y} = \b{0}}$$
Notice that,
$$T(\b{x}^*) = \nullspace{ D \b{h}(\b{x}^*)}$$
The tangent space is therefore a subspace of $\R^n $.
Assuming that $\b{x}^* $ is regular, the dimension of the tangent space is $n - m$, where $m $ is the number of equality constraints $h_i(\b{x}^*) = 0$.
Note that the tangent space passes through the origin.
The tangent plane at $\b{x}^* $ to be the set
$$TP(\b{x}^*) = T(\b{x}^*) + \b{x}^* = \set{ \b{x} + \b{x}^* : \b{x} \in T(\b{x}^*)}$$
$S = \set{ \b{x} \in \R^n : \b{h}(\b{x}) = \b{0}} $.
Suppose that $\b{x}^* \in S$ is a regular point and $T(\b{x}^*) $ is the tangent space at $\b{x}^* $.
$\b{y} \in T(\b{x}^*) \iff$ there exists a differentiable curve in $S $ passing through $\b{x}^* $ with derivative $\b{y} $ at $\b{x}^* $.
The normal space $N(\b{x}^*) $ at a point $\b{x}^* $ on the constraint surface $S$ is the set
$$N(\b{x}^*) = \set{ \b{x} \in \R^n : \b{x} = D \transpose{ \b{h}(\b{x}^*)} \b{z}, \b{z} \in \R^m}$$
Notice that,
$$ N(\b{x}^*) = \range{ \transpose{ D \b{h}(\b{x}^*)}} $$
The normal plane at $\b{x}^* $ to be the set
$$NP(\b{x}^*) = N(\b{x}^*) + \b{x}^* = \set{ \b{x} + \b{x}^* : \b{x} \in N(\b{x}^*)}$$