What does surface integral define? What does a surface integral define.... what is the difference between $dS$ and $dA$ if any..?
I know that $dA$ give the volume of an function with 2 variables, but what about $dS$ does it do the same or what..?
Trying to study using this material
 A: These are just notations that confuses people a little. That is why I greatly prefer to write the integral of any function $f : A \subset \Bbb R^n \to \Bbb R$ simply as:
$$\int_A f$$
But here comes what this notation is meant to tell you: we usually use $dA$ to say that we are integrating over some subset of $\Bbb R^2$. Now, a surface can be thought of as the resulf of taking pieces of the plane, bending then, and putting it all together in a smooth manner. When we integrate over a surface we usually use $dS$ to indicate that.
In the old and informal way of thinking about these, it was all about really small patches of area. So the idea is that $dA$ would be an infinitesimal patch of area in the plane and $dS$ an infinitesimal patch of area on some arbitrary surface. These notions however, tend to just complicate things instead of simplify, so the modern treatment doesn't use this anymore.
Just as curiosity, these $dS$ and $dA$ only make sense when you go to Differential Geometry. In this context, these will be the "volume forms" of the manifold. So $dA$ would be the "standard notation" for the volume form in the plane and $dS$ would be some "standard notation" for the volume form in arbitrary surfaces. And of course, "volume" in this context is just the area.
A: $A$ stands for Area and an integral versus $dA$ is an integral over an area - that is plane.
$S$ stands for Surface and an integral versus $dS$ is an integral over surface itself - that can be curved or plane.
The confusing part is often comming from the fact that the projection of a curved surface on the carteisan coordinates' planes e.g. $xy$ or $xz$ or $yz$ is an area.
Extension:
Regarding your second comment/question I think what may be confuses you is the term "volume". Indeed height plays a role beacuse a surface when curved curves in a volume. But a volume can be also an area as in DG. In order to understand this suggest you have a toy experiment: take a piece of paper and build a surface then examine $dS$ as well $dA$ visually. For $S$ the paper curved in space and for $A$ plane. Then try to find ways to put them in relation for instance $A$ could be plane projectin of $S$ or an infitessimal piece of $S$ (just mark a smal piece with a pen) can be approximated by $A$... Then imagine to walk on the $S$ and sum up all you see on the landscape ($u$,$v$) and the same when walking on $A$. It will give you an intution of what is the difference.
Hope this helps you more visually.
