What does $f(x,y)$ mean? I know from the chapter "functions" that $f(x)$ is a function of $x$ and to roughly put it, it maps $x$ values to another set called co-domain where all the $y$ values are.
But I also sometimes see $f(x,y)$ on internet. I can guess that it means some expression in $x$ and $y$.
I'm not familiar with them yet and they aren't in my high school syllabus but I'd love to know more about and I have a few questions,

  
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*What type of function is this? What's it called?
  
*What does is represent? Can you also represent $f(x,y)$ using arrow diagram between $2$ sets?
  
*Is $(x,y)$ in $f(x,y)$ an ordered pair? Or $f(x,y)$ is same as writing $f(y,x)$


Here is a short answer to your questions:
1) The function can be called a bivariate function; it is a function that depends on two variables $x$ and $y$ that may assume different domains. The function is defined on the union of those domains. An example is 
$$ f(x,y) : = x^2 + y^2$$
If you fix $x$ to any value say $\bar{x}$, then $f(\bar{x}, y)$ is a function in $y$. The same holds if you fix $y$ instead, then the function becomes a function in $x$.
2) it represents a rule of mapping values of $x$ and $y$; you can still use arrow diagrams yes.
3) When you define the function, the pair $(x,y)$ should be ordered. But it is a matter if notation which argument u want to appear first.
