You can call f(x,y) might be called a function of two variables. An example that might be helpful would be something like f(x,y)=x2+y2. This function takes in two numbers, takes their squares and adds them. Note that a function of two variables doesn't need to do the same thing to both variables. We could also make a function g(x,y)=x3+xy+1 for example. You can represent them using an arrow diagram. Your set to the left of your arrow will be the set of ordered pairs of (x,y). And yes, the you do need to think of (x,y) as an ordered pair. It is not necessarily the case that f(x,y)=f(y,x). See the g(x,y) example above and note that g(0,1)=1 but g(1,0)=2.
Many functions in real life are functions of more than one variable. An example from physics: the gravitational pull an object experiences from a planet is a function of both the planet's mass and the distance to the planet.