Suppose $Nul(A)$ has dim $> 0$. This means there are infinite solutions to the normal equations $A^{T}A\hat{x} = A^{T}b$. That is, you can define $\hat{x}$ parametrically.
From what I understand about the least-squares solution $A\hat{x}$, it is the closest point from the $Col(A)$ to $b.$ But what does it look like when there are infinitely many closest points? I'm having a hard time trying to visualize it.