What does it mean for a level curve to be closed or open? So, I understand what it means to be closed/open or bounded/unbounded in terms of domain.
What about level curves? For example, for $f(x,y) = x^2 - y^2 = c$, is this closed/open or bounded/unbounded?
I thought these terms refer to a region, not a line or curve. 
How could we determine if a level curve is open or closed and boudned/unbounded?
 A: A closed curve is by definition a continuous image of a circle.  This is not the same meaning of closed as in "closed set". In particular a closed curve is bounded.
A level curve $f(x,y) = c$ of a smooth, nowhere constant function, if it is bounded, typically consists of one or more closed curves. 
A sufficient condition for $f(x,y) = c$ to be bounded is that $|f(x,y)| \to \infty$ as $|x| + |y| \to \infty$.  
A: The level curve is a subset of the domain. It's perhaps not what you would typically consider a "region" since it does not have the same dimension as the domain, but the definitions of being an open or closed set (every point is an interior point; contains all of its limit point) can still be applied. For your function $f(x,y)=x^2-y^2$ it's clear that all of the level sets are closed and not open; in fact it can be proven that the level sets of all continuous functions are closed (this is sometimes used as the definition of a continuous function).
I don't know that there's an easy test for whether a given function's level curve is bounded, other than a case-by-case analysis. There are some sufficient conditions which guarantee that all level curves will be bounded; for example, if the function is strictly convex.
