What's function $f(x)=x$ like? I didn't understand the function $f(x)=x$. Does it mean that output is equal to input? What about its graph, what does it look like?
 A: You are absolutely right. 
It means that for all $x$, the output $f(x)$ is the same as the input $x$.
It also means geometrically that if you interchange the $x$-axis and the $y$-axis, you won't change the graph of the function:

A: Other answers give very good explanation for x in the real line. I'll try to give a more abstract explanation. 
To be strict, $f(x) = x$ itself does NOT define what x is, and we could say x is an element of a set/space. (And one natural concrete example would be a point x $\in \mathbb R$).
But in general, we could say $x \in \Omega$, where $\Omega$ is an arbitrary set. And $f : \Omega \mapsto \Omega$ is a map that maps every element of $\Omega$ to itself. And BTW, this map is a bijiection.

A: It is the identity function and yes the input is equal to the output.
Its graph is the straight line passing through the points $(0,0)$ and $(1,1)$.
A: Each and everytime you throw an input to it your output is the exact same thing
What this means is if you plot the graph with y axis as your output and x axis showing your input.
For a unit change in x there will always be the same the change in y. In short no matter which two points you decide to find the slope between that slope always remains consistent.
So the graph looks like a straight line passing through the origin for real input and output values.
It is called the identity function as it returns the identity of the parameter given to it.
A: This function represents a straight line passing through origin and having slope = 1.
