Let $A = k[x]$ where $k$ is a field. Then an $A$-module is just a $k$-vector space $V$ equipped with a $k$-linear map $\widehat{x}: V \to V$.
One point of confusion with this is as follows. So is a specific $A$-module equipped with a fixed linear map $\widehat{x}$ that corresponds to a specific element in $k[x]$? Or is $\widehat{x}$ allowed to vary as to correspond to all elements of $k[x]$? Or am I thinking of this in the wrong way and does this $\hat{x}$ correspond to $k[x]$ aggregate and not an element or elements in it?
Can anyone provide me an example as to clarify my confusion?
The $\hat{x}$ represents one particular linear map $V \to V$. Since $A = k[x]$, you can express every element of $A$ as a $k$-linear combination of powers of $x$. Now, the vector space $V$ is an $A$-module precisely when you have a linear map $V \to V$ associated to every element of $A$. Since you know how to do this for the element $x$ (namely, the associated map is $\hat{x}$), you can associate to a polynomial $p(x) \in A$ the map $p(\hat{x})$.