How do we know if a subset S of R^2 represents an area? As an example:
Let $S_1 := \{ (x,y) | x^2 + y^2 \leq 1 \} \Longrightarrow S_1 \subset R^2$,
which represents the area of a "filled" circle with radius $r = 1$ and center $c = (0,0)$. What exactly makes $S_1$ an area? How do we know that this is not just a "random" collection of points, such as for example $S_2 = \{(0,0), (1,1), (2,2)\}$? Intuitively, I would say a subset $S$ is an area, if $Area(S) > 0$, but then, how would you define the $Area$ function?
 A: I'm wondering where you are seeing this use of the word "area", in the assertion that the set $S_1$ "represents the area of a "filled" circle". It sounds like an intuitive notion that has no strict mathematical definition, so I doubt that your question "what exactly makes $S_1$ an area" is answerable.
As you say, generally speaking "area" is a function which assigns a non-negative real number to a certain collection of subsets of $\mathbb R^2$. In analysis one learns that the subsets in this collection are known as the measurable subsets. To put it another way, the measurable subsets $A \subset \mathbb R^2$ are exactly those subsets to which a value of $\text{area}(A) \in [0,\infty)$ is assigned. This "measurable" property is a true--false property: some subsets are measurable; others are not. You can see this theory developed in many analysis textbooks (see the links in the comment of @JMoravitz).
So, for example, the set $S_1$ in your question is measurable, and its area is $\pi$.
Also, your set $S_2$ is measurable, and its area is zero.
But I have not ever seen "area" used itself as a true--false property of subsets of $\mathbb R^2$, i.e. one does not say "this subset of $\mathbb R^2$ is an area; that subset is not".
