I have read that a fiber is the pre image of a mapping..
Does this mean that I can think of a fiber as a line that connects x to y where the line is a function?
So for example with $f(x)=2x$ there is a line going from $x$ to $2x$ for all $x$ and these are all the fibers? Is that right?
The fiber of, say, $y=1$ is the set of all $x$ such that $f(x)=1$.
For your example, $f(x)=2x$, the fiber of $1$ is the set $\left\{\dfrac12\right\}$, the fiber of $8$ is $\{4\}$, etc.
More interesting examples when the function is not 1-1, like take $g(x)=x^2$. Then the fiber of $4$ is $\{-2,2\}$, the fiber of $16$ is $\{-4,4\}$.
You use the word pre-image: The pre-image (or fiber) of a point $y$ is the set $f^{-1}(y):=\{x:f(x)=y\}$.