Visual meaning of simple surfaces in Differential Geometry According to Lipschutz's Differential Geometry book, a simple surface is deffined this way:

Let $S$ be a set of points in $E^{3}$ for which there exists a collection $\mathcal{B}$ of coordinate patches of class $C^{m}(m \geqslant 1)$ on $S$ satisfying

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*$\mathcal{B}$ covers $S$, i.e. for every point $P$ in $S$ there exists a coordinate patch $\mathbf{x}=\mathbf{x}(u, v)$ in $\mathcal{B}$ containing $\boldsymbol{P}$

*Every coordinate patch $\mathbf{x}=\mathbf{x}(u, v)$ in $\mathcal{B}$ is the intersection of an open set $O$ in $E^{3}$ with $S$.

Then $S$ together with the totality of coordinate patches of class $C^{m}$ in $S$ is a simple surface of class $C^{m}$ in $E^{3}$.

I wonder what is the visual meaning of a surface that is simple and a surface that is not simple. In the case of curves, a curve was said to be simple if it does not cross itself. Do simple surfaces have a similar meaning?
 A: Lipschutz's definition of a simple surface is equivalent to the definition of a regular surface in doCarmo's book Differential Geometry of Curves and Surfaces. The latter might be a better starting point to learn differential geometry with a more standard & modern terminology.
In chapter 5.10 of DoCarmo's book he compares several further notions of surfaces, starting with that of an abstract surface, which is the same thing as a smooth $2$-dimensional manifold. In summary, as elaborated on on page 441:

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*An immersed surface is the image of an abstract surface under an immersion

*A regular surface is the image of an abstract surface under an embedding. An embedding is a map that is both an immersion and and a homeomorphism onto its image.

The notion of immersion/embeddings is modern terminology and essentially boils down to allowing self-intersections (for immersions) and not allowing them (for embeddings). So e.g. the typical picture of a Klein bottle you may be familar with is an immersed surface  (it has self-intersections) but not an embedded surface. It does not meet Lipschutz's second criterion (2), because at a self-intersection, a particular coordinate patch will only cover one branch, while the intersection of open subset with the surface must contain parts of both branches.
As a sidenote: There is also a more modern usage of the term simple surface, which is a lot more restrictive. (It is a Riemannian  surface with strictly convex boundary, such that any two points are connected by a unique geodesic, depending smoothly on the endpoints).
