Is a function a special kind of relation? Is a function a "special kind of relation", or, does it "describe a specific relation"?
My text on discrete mathematics explains:

A relation is a subset of a Cartesian product and a function is a
  special kind of relation.

But it would make more sense to me if a function described a relation as a subset of the Cartesian product.
My thoughts being:
Given a function, f(x) = y, we can compute a set of (x,y) coordinates within the Cartesian plain. And this set of coordinates would be the relation that is the subset of the Cartesian product.
Am I confused? Could anyone help explain how a function IS a relation?

Think of the relation "is less than" in $\{1,2,3\}^2$. That relation is $\mathrm{Less}=\{(1,2),(1,3),(2,3)\}$, however you normally will not write $(x,y)\in\mathrm{Less}$, but you will write $x<y$. You certainly have no problem with saying "$x<y$ is a relation". And of course that is true not only for $x<y$, but also for other relations like $x\le y$, $x<y^2$, $x=y$, or, for that matter, $f(x)=y$.
So we see, $f(x)=y$ is a relation. But as you correctly said, $f(x)=y$ is a function. So a function quite obviously is a relation.
It is, however, a very special relation; which is why the notation $f(x)$ makes sense: For any $x\in A$ there's exactly one $y\in B$ so that $(x,y)\in f$. That is, you can read "$f(x)$ as meaning "The unique $y$ such that $(x,y)\in f$", just as you can read $xRy$ as "$(x,y)\in R$". Note that this also means that in principle, $xfy$ would also be a meaningful way to write $f(x)=y$, although that notation is, well, extremely unusual.
