Two types of surfaces in $\mathbb{R}^3$ are usually studied in introductory books on differential geometry:
Parametrised or immersed surface: Is an immersion $F:U\rightarrow\mathbb{R}^3$ from an open set $U\subset\mathbb{R}^2$ (the differential $\text{d}F(u):\mathbb{R}^2\rightarrow\mathbb{R}^3$ is injective on $U$). The trace or image $S=F(U)\subset\mathbb{R}^3$ of the immersion is often called the parametrised surface. Parametrised surfaces are studied in Kuhnel Differential Geometry: Curves-Surfaces-Manifolds and in higher dimensions (parametrised hypersurfaces) in the literature on mean curvature flow.
Remark: Occasionally a parametrised surface is defined to be an injective immersion, in which case the trace $S=F(U)\subset\mathbb{R}^3$ becomes an immersed submanifold (itself a $2$-dimensional manifold for which the inclusion $\iota:S\hookrightarrow\mathbb{R}^3$ is an immersion).
Regular or embedded Surface: Is a subset $S\subset\mathbb{R}^3$ for which every point $p\in S$ has a local parametrisation, namely a smooth homeomorphism $\mathbf{x}:U\rightarrow S\cap V$, where $U\subset\mathbb{R}^2$ is an open set and $V\subset\mathbb{R}^3$ is an open neighbourhood of $p$, whose differential $\text{d}\mathbf{x}(u):\mathbb{R}^2\rightarrow\mathbb{R}^3$ is injective on $U$. Regular surfaces are studied in do Carmo Differential Geometry of Curves and Surfaces.
Remark: A regular surface is just a $2$-dimensional regular submanifold $S\subset\mathbb{R}^3$, namely a $2$-dimensional manifold with the subspace topology for which the inclusion $\iota:S\hookrightarrow\mathbb{R}^3$ is an embedding (an immersion which is also a homeomorphism onto its image).
We can show that regular surfaces are particular instances of parametrised surfaces: Regular surfaces are the image of an embedding (the inclusion map), and conversely the image of an embedding (between $\mathbb{R}^2$ and $\mathbb{R}^3$) is an embedded surface.
Question 1: What is the geometric intuition behind the difference between parametrised and regular surfaces?
Question 2: What are some examples of parametrised surfaces that are not regular surfaces? (Please prove your example is not a regular surface.)
Best Answer
All of my geometric intuition for "immersed" versus "embedded" surface is contained in my geometric intuition for "immersions" versus "embeddings". In particular, as many people have pointed out, immersions need not be injective. But, of course, even injective immersions need not be embeddings. As a very simple example, consider the map $f : (-\pi, \pi) \to \mathbb{R}^2$ given by $$ f(t) = (\sin t, \sin 2t). $$ The image of this map is a sort of "figure 8" in the plane, traced out starting at the origin, moving through quadrants II, III, I, and IV, in that order, as $t$ moves from $-\pi$ to $\pi$. It's easy to see that $f$ is an injective immersion, but $f$ is not an embedding, since every neighborhood of $\mathbb{R}^2$ containing the origin also contains points of the form $f(-\pi + \epsilon)$ and $f(\pi - \epsilon)$ for all sufficiently small $\epsilon$.
As another one-dimensional example of this type, you could consider the closed topologist's sine curve with a loop, which is the graph of $g(x) = \sin(1/x)$ for $x \in (0, 1]$ together with the $y$-axis between $y = -1$ and $y = 1$ together with a "loop" smoothly connecting the point $(0, -1)$ to the point $(1, \sin(1))$. It's clear that there is some injective immersion $f : [0, \infty) \to \mathbb{R}^2$ whose image is this curve, and this immersion is not an embedding.
You can, of course, easily make either of these example into a surface by considering $h: (-\pi, \pi) \times (0,1) \to \mathbb{R}^3$ given by $h(t, s) = (f(t), s)$.
On the other hand, I don't really know any examples of injective immersions of surfaces which aren't embeddings that are "interesting" in a way that's fundamentally different from the examples above. The idea I have is that immersions are allowed to "approach themselves" or "limit onto themselves" in crazy ways that embeddings are not. In particular, if $f : X \to Y$ is an injective immersion, the topology on $f(X)$ as a subspace of $Y$ might be very different than the topology on $X$.