A local system is a sheaf of finite dimensional vector spaces that is locally isomorphic to the constant sheaf $k^n$. If $\gamma: [0,1] \to X$ is a continuous path in $X$, $\gamma^{-1}(L)$ is again local system on $[0,1]$ but one shows that any locally constant sheaf on $[0,1]$ is actually constant. So the fibers at 0 and 1 are canonically identified. This means that we have a map
$$
\gamma_{*} : L_{\gamma(0)} \to L_{\gamma(1)}
$$

Moreover this is

- linear: $\gamma_* (v + \lambda w) = \gamma_* (v) + \lambda \gamma_* (w)$
- invariant by homotopy: if $\gamma \sim \gamma'$, $\gamma_*v = \gamma'_*v$.
- compatible with composition of homotopy classes of paths: $(\gamma')_*(\gamma_*x) = (\gamma'\gamma)_*x$.

So to any local system $L$ corresponds a representation $\pi_1(X,x) \to GL(L_x)$ of the fundamental group at $x$. This is the monodromy representation. You can rebuild $L$ from it: this is the sheaf of sections of $(\tilde{X}\times V)/\pi_1(X,x) \to X$ where $\tilde{X}$ is the universal covering of $X$. We have sketched:

**Theorem**: If $X$ is connected, the functor "fiber at $x$" induces an equivalence of categories $LS(X) \to Rep(\pi_1(X,x))$.

This is all very abstract so let us look at an example.

Consider $\mathcal{K}$, the trivial rank 2 vector bundle $\cal{O}_{\mathbb{C}^\times}^2$ and connection
$$
\nabla \begin{pmatrix} f_1 \cr f_2 \end{pmatrix} =
d\begin{pmatrix} f_1 \cr f_2 \end{pmatrix} -
\begin{pmatrix}0 & 0 \cr 1 &0 \end{pmatrix} \begin{pmatrix} f_1 \cr f_2 \end{pmatrix} \frac{dz}{z}
= \begin{pmatrix} df_1 \cr df_2 - f_1 \frac{dz}{z} \end{pmatrix}
$$

Horizontal sections are the solutions of $\nabla f = 0$. This is a system of two first order linear differential equations. On any simply connected $U$ we can chose a determination of the logarithm and the solution can be written

$$
\begin{pmatrix} f_1 \cr f_2 \end{pmatrix} =
\begin{pmatrix} A \cr A \log z + B \end{pmatrix}
$$
where $\log$ is any determination of the logarithm function. This means that the sections
$$
e_1 = \begin{pmatrix} 1 \cr \log z \end{pmatrix} \qquad
e_0 = \begin{pmatrix} 0 \cr 1\end{pmatrix}
$$
trivialize the sheaf of solutions on $U$. Covering $\mathbb{C}^\times$ by simply connected open sets we see that the solutions form a local system $L$.

When we turn once around 0 following the orientation, our determination $\log z$ changes to $\log z + 2\pi i$. So if $\gamma(t) = xe^{2\pi i t}$

$$
v = \begin{pmatrix} A \cr A\log x + B \end{pmatrix} \mapsto
\begin{pmatrix} A \cr A(\log x + 2\pi i) + B \end{pmatrix} = \gamma_*(v)
$$
The monodromy representation is
$$
\pi_1(\mathbb{C}^\times,x) = \mathbb{Z} \to GL_2(\mathbb{C}) \qquad
1 \mapsto \begin{pmatrix} 1 & 0 \cr 2\pi i & 1 \end{pmatrix}
$$
It tells you everything there is to know about our differential equation (because it has regular singularities). For example, the space of global solutions is identified with the space of invariant of the representation:
$$
\Gamma(\mathbb{C}^\times,L) = Hom(k_{\mathbb{C}^\times},L) = Hom_{\pi_1(X,x)}(k,L_x) = L_x^{\pi_1(X,x)}
$$
This is the 1 dimensional space generated by $e_0$.

Another good example to work out is the equation $df = \alpha f\frac{dz}{z}$ (the monodromy can be quite different depending on $\alpha$).

Pierre Schapira's webpage has notes of a course on sheaves and algebraic topology focussing on local systems. Claire Voisin's book on Hodge theory is a good reference for variations of Hodge structures.