Suppose $X$ is a pointed and connected topological space and $L$ is a local system on $X$. By this I mean that $F$ is a locally constant sheaf on $X$, or equivalently, a representation of the fundamental group $\pi_{1}(X,x)$ of $X$.
A classical example of a local system is the sheaf
$$ U\mapsto L(U) = \{f:U\to \mathbb{C}\mid \mathcal{D}(f) = 0\} $$
of complex solutions to a system of differential operators $\mathcal{D}$ on $X$. This generalises to the sheaf of horizontal sections of a vector bundle with flat connection $(V,\nabla)$ on $X$. In fact, all local systems on $X$ arise in this way.
Given a local system $L$ on $X$, one can define cohomology groups $H^{n}(X,L)$ of $X$ with coefficients in $L$. Sadly, I only understand this theory in a formal way. The aim of my question is to understand cohomology with coefficients in a local system in elementary differential geometric terms. To this end I have two questions:
Question 1: What is a geometric interpretation of cohomology classes $\alpha\in H^{n}(X,L)$ when $L$ is interpreted as a vector bundle with flat connection?
Question 2: In terms on solutions to differential equations, what does it mean for cohomology to vanish in degree $n$? What does it mean for a map of local systems to induce the zero map on cohomology?
I don't have an answer for twisted complex systems, but if $\Bbb Z_w$ denotes the twisted system with twisting $w: \pi_1 X \to \{\pm 1\}$, then $X$ comes equipped with a canonical real line bundle $\Bbb R_w$, and geometric representatives for $H_*(X;\Bbb Z_w)$ are given by manifolds $M$ equipped with a map $f: M \to X$ and an isomorphism $\det(TM) \cong f^*\Bbb R_w$. This is called a twisted orientation.
This is useful when trying to geometrically understand Poincare duality on a non orientable manifold.