So the definition of a regular surface is the following:
A topological subspace $S \subset \mathbb{R^3}$ is a regular surface if $\forall p \in S$ there exist open sets $U \subset \mathbb{R^2}$, $V \subset S, p \in V$ and a function $\phi:U \rightarrow \mathbb{R^3}$ with $\phi \in C^{\infty}(U, \mathbb{R^3})$ such that:
$\phi:U \rightarrow V$ is a homeomorphism.
$d\phi_q:\mathbb{R^2} \rightarrow \mathbb{R^3}$ is injective $\forall q \in U$ ($d\phi_q$ is the differential of $\phi$ in q).
Regular surfaces cannot have self-intersections. I guess this has to follow from the definition of regular surface given above, probably the condition of $\phi$ being an homeomorphism, but I can' t see exactly why.
Assume that $p\in S$ is a point of self-intersection of $S$, then there exists $x_1\neq x_2$ in $U$ such that: $$\phi(x_1)=\phi(x_2),$$ contradicting the injectivity of $\phi$.
Let us discuss the geometrical/physical meaning of each of the conditions:
A point of $S$ is tracked with coherent local coordinates $(x,y)$, in some geographical sense, $S$ is charted. Said differently, you have a grid on $S$.
The speed of a particle moving on $S$ is well-defined; in other words, $S$ does not have corners nor sharp edges, it is a smooth object.