What's the difference between a curve and a graph? My book states "... if $S$ is a level curve and $C$ is a curve in $S$ passing through a point $a$ ..."
What is a curve in $S$? Is it simply a subset of $S$?
From what I comprehend, a graph is usually a ordered set of the form 
$ X = \{(a,f(a))\ |\ a \in \mathrm{dom}(f)\}$
 A: By "curve in $S$" your book means a restriction of $S$ which still represents a curve. For example, if we consider the curve described by
$$
(x,y) = (\sin\theta, \cos\theta)
$$
for $\theta \in [0, 2\pi]$, then an example of a curve "inside" this curve would be
$$
(x,y) = (\sin\theta, \cos\theta)
$$
for $\theta \in [0, \pi/2]$.
Probably the reason your book worded it this way is because it doesn't want to include every such restriction. For instance, restricting the domain to $[0,2\pi ] \cap \mathbb{Q}$ gives a subset of the curve, but not a "subcurve" (this terminology is not used as far as I know, but I didn't know what else to call it.)
Note that in introductory calculus courses often times there is no distinction made between a curve, the range of a curve, the graph of a curve, and the parametric representations of a curve. This is often times a source of confusion (as it may have contributed here), but the distinction should be properly handled in a more rigorous analysis setting.
You are correct that the graph of a curve is given by $\{(a,f(a)) | a \in \mathrm{dom}(f)\}$
