Let $X$ be a complex smooth manifold. The data of a rank-$n$ local system (a sheaf on $X$ that is locally the constant sheaf $\underline{\mathbb{C}^n}$) is the same thing as an open cover $\{U_\alpha\}$ of $X$ and a choice of elements $g_{\alpha \beta} \in GL_n(\mathbb{C})$ telling you how to identify the stalks on overlap. In other words, we get a $\mathbb{C}$-vector bundle of rank $n$ that admit an open cover whose transition functions are constant. The latter object is often identified with a vector bundle with a flat connection. And I want to make this identification clear because a lot of the arguments I have read simply mumble this through and hence is not down-to-earth enough for me to understand.
Let's fix some terminology (as there seems to be different approaches to these stuff):
A connection on a vector bundle $E \to X$ is a (non $C^\infty(X)$-linear) map $$\nabla: \Gamma(E) \to \Gamma(T^*_\mathbb{C} X) \otimes_{C^\infty(X)} \Gamma(E)$$ satisfying the Leibniz's rule $\nabla (f \sigma) = f \nabla \sigma + (df) \otimes \sigma$. (By thinking of $\otimes \Gamma(E)$ as extension of scalar, $\nabla \sigma$ can be regarded as an $E$-valued 1-form.) The curvature of $\nabla$ is defined by the composition of $\nabla$ with the map $$\tilde{\nabla}: \Gamma(T^*_\mathbb{C} X) \otimes_{C^\infty(X)} \Gamma(E) \to \Gamma(T^*_\mathbb{C} X) \otimes_{C^\infty(X)} \Gamma(T^*_\mathbb{C} X) \otimes_{C^\infty(X)} \Gamma(E)$$ defined by $\tilde{\nabla}(\theta \otimes \sigma) = d\theta \otimes \sigma - \theta \otimes \nabla \sigma$. We say that a connection is flat if its curvature is $0$.
(I have only learned these stuff very recently, and so to this date, I still don't know what is the curvature trying to accomplish. A connection gives a way to differentiate sections of $E$, that's all the intuition I have now.)
In the sense defined above:
(1) Given a vector bundle with open cover with constant transition functions, how do I define a flat connection?
(2) Conversely, given a flat connection on a rank-$n$ vector bundle, how do I construct such a nice open cover?