$A$ is the power set of set $C$, and $S$ is the relation on $A$ defined by $x S y$ iff $x \subseteq y$. The following is an exercise from my textbook:

Let $C$ be a set and let $A=\mathcal{P}(C)$, the set of all subsets of $C$. Let $S$ be the relation on $A$ defined by $x S y$ iff $x \subseteq y$.  Then $S$ is a partial order relation on $A$.
  (a) Find the least and greatest element of $A$.
  (b) Let $B=\{x \in A$: $x \ne ∅\}$.  Suppose $C$ has two or more elements.  Show that $B$ has no least element.   

I'm confused about the definition of least and greatest element as applied to this exercise.   
From the my textbook, least element is defined as such:   

Let $(A,\leq)$ be a partially ordered set and let $B$ be a subset of $A$.  To say that $c$ is a least element of $B$ means $c \in B$ and for each $x \in B$, $c \le x$.   

Now, referring back to the exercise, $A$ consists of sets, not merely numbers; thus, to say that some set $x_1$ is less than or equal to some other set $x_2$ doesn't make sense.  We don't relate sets in that way do we?  (Unless, we're talking about order?  But even so, I don't think the exercise is asking us to consider that.)
Now, part (b) leads me to believe that a least element of some set of sets $D$, if it exists, is some set such that it is a subset of each set in $D$.  Is that correct?
 A: The notation of $\leq$ is often used in mathematics as a general partial order. Not just the usual ordering on the natural/integers/rationals/reals. But in this context, $\leq$ is really just $\subseteq$.
So you are being asked to find the maximum and minimum elements of $(\mathcal P(C),\subseteq)$.
A: In your case, the least element is an element of $A$, hence a subset of $C$ that is contained in all subsets of $C$. (Apply your definition in the second highlighted paragraph with $A=\mathcal{P}(C)$ and $B=A$), and so this set, I guess, is the empty set. Now, for the greatest element of $A$, what is the subset of $A$ that contains  any subset of $A$?
A: Okay, I know that what follows basically repeats what responders have been trying to tell me, but hitherto, I was still misunderstanding the notation of $\le$.  Thus, I merely post this answer as an expression of my understanding (finally!).
I refer to the definiton:   

Let $(A,\le)$ be a partially ordered set and let $B$ be a subset of $A$.  To say that $c$ is a least element of $B$ means $c \in B$ and for each $x \in B$, $c \le x$.   

The $\le$ in the ordered pair $(A,\le)$ does not refer to the relation of less than or equal to, but rather serves as a "variable" for some partial order relation on A.  With this in mind, the condition for each $x \in B$, $c \le x$ doesn't mean that $c$ must be less than or equal to $x$, but rather that $c$ must be in relation $\le$ to $x$.   
Now, the partial order relation specified in the above exercise is the relation of set inclusion.  Then, by definition, the least element of $A$ must be some set in $A$ such that it is a subset of each set in $A$.
