What does it mean for a point to be "inside" a plane curve? Suppose that we have a simple closed curve in the $xy$-plane: $f(x,y)=0$ (for example, $f(x,y) = x^2 + y^2 - 4$. If I give you the point $(x,y)$, is there an analytical method to determine whether $(x,y)$ is "inside" $f$? (besides looking at a graph, etc.)
That raises the following question: what exactly does it mean to be "in" a curve? The closest definition my friends and I achieved involved a new space where the line integral of the curve from a given point to a second point is mapped to the distance from that second point to $(x,y)$.
Of course, I feel like this must have been solved a long time ago, and my explanation was rather naïve. Still, I'm having a hard time navigating Wikipedia and the such (MRH).
What does it mean to be inside a simple closed curve?
 A: The winding number will be $1$ inside a simple closed curve and zero outside.
If you have a well-behaved simple closed curve imagine a line (ray i.e. one direction) from your point to infinity. If the line crosses the curve an odd number of times the point is inside the curve. 
For polygons see, for example, this link which deals with both ideas.
A: Assume your simple closed curve is the zero set of a function
$$ C: f(x,y) = 0$$
and $f$ also takes positive and negative values. (thanks @Federico Poloni for drawing attention to the need of this last condition).
Now the complement of the curve has two connected components by Jordan's theorem so none of these components can intersects both of the non-void sets $\{f<0\}$ and $\{f>0\}$. It follows that the components of $\mathbb{R}^2 \backslash C$ are  $\{f<0\}$  and $\{f>0\}$. Now you have to figure out what is the bounded component (the inside). Look at the sign of $f$ for large values of $(x,y)$ ( at infinity). The inside is the region of opposite sign. 
Inside means not with the infinite. 
