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 $f(y,x)$


That's called a function of two variables. A function $f:\mathbb{R^2} \to \mathbb{R}$ is a  function that maps a pair of real numbers $(x,y)$ to another real number $z$. Of course we don't limit ourselves only to two variables, the general case are functions from $\mathbb{R^n}$ to $\mathbb{R^m}$, they map $n$-tuples $(x_1,...,x_n)$ to $m$-tuples $(y_1,...,y_m)$. It's above high school level though. 
