If you have two sets $A$ and $B$ where $A=B=\{\;\}$, then what would $A\times B$ be equal to?
I know that $A\times B=\{(a,b)|a\text{ is an element of }A\text{ and }b\text{ is an element of }B\}$.
I think I am getting confused as to the notation of an element of an empty set or even if you can have one.
Keep calm when you get confused, just apply the definition.
$A\times B$ should consist of exactly those $(a,b)$ such that $a\in A$ and $b\in B$. Now if any of $A$ or $B$ is empty that simply doesn't happen and therefore there is no $(a,b)$ that qualifies for being a member of $A\times B$, that is there's no member of $A\times B$. This means that we always have
$$\emptyset\times X = \emptyset$$ $$X\times\emptyset = \emptyset$$
for any set $X$. Especially we get:
$$\emptyset\times\emptyset = \emptyset$$