I can interpret all this definition except the $b)$.
Help me with visual examples what's its interpretation.
$\mathbf{Definition} $: A topological sub-space $S\subset\Bbb R^3$ is a regular surface when for all points $p\in S$, there are open sets $U\subset\Bbb R^2$, $V\subset S$ with $p\in V$ and $\varphi\in C^{\infty}$, $\varphi:U\to\Bbb R^3$ which verifies:
$a)\;\varphi:U\to V$ is a homeomorphism.
$b)\;d_{\varphi_{q}}:\Bbb R^2 \to \Bbb R^3$ is injective for all $q\in U$, where $d_{\varphi_{q}}$ is the differential of $\varphi$ in the point $q$.
Imagine the surface of a cone. Every point of this surface satisfies the required condition, except for the tip of the cone. For the tip $p$ there exist $U,V,\phi$ satisfying the required conditions, except for (b) -- there does not exist a homeomorphism from a neighborhood of $0\in \mathbb R^2$ to a neighborhood of the tip $p\in S$ whose derivative in $0$ is injective. There might be one whose derivative in $0$ is non-injective. To see why, imagine the 1-dimensional analogue of situation, where the cone is replaced by the curve $C=\{(x,|x|): x\in\mathbb R\}\subset \mathbb R^2$, where $|x|$ is the absolute value. Then the function $f$ such that $f(x)= (x^3,|x|^3)$ is a homeomorphism from $\mathbb R$ to $C$ which maps $0$ to $(0,0)\in C$, but its derivative at $0$ is equal to $0$, hence is non-injective. It is not difficult to replace this homeomorphism $f$ by one which is $C^\infty$, with similar properties.