TITLE: Why are local systems and representations of the fundamental group equivalent
QUESTION [44 upvotes]: My question: Let X be a sufficiently 'nice' topological space. Then there is an equivalence between representations of the fundamental group of X and local systems on X, i.e. sheaves on X locally isomorphic to a constant sheaf. Does anyone know of a self contained, detailed treatment of this suitable for my background? I've looked at the first few pages of Delignes "Équations différentielles à points singuliers réguliers" (which my advisor suggested I take a look at) but here it just says that the equivalence is "well known", giving no reference. Neither googling ("local systems representations fundamental group") (nothing usable comes up), wiki nor the nLab entry (not detailed anough and more interested in generalisation) on local systems were of much help to me. I apologise in case the equivalence should obvious once one knows about universal covering spaces/deck transformations. I haven't learned those yet. If so, please let me know. 
Why I care: I am trying to read Simpsons "Higgs bundles and local systems",. Publ. Math. I. H. E. S. 75 (1992) 5–95". Simpson assumes this equivalence but gives no references. 
If there is any way this question could be improved upon, please let me know. 
Feel free to retag.

REPLY [4 votes]: I'd strongly recommend having a look at Atiyah & Bott's paper The Yang-Mills Equations over Riemann Surfaces.  They explain this pretty well, and embed it in a beautiful larger picture.  
The basic idea is that a locally constant sheaf can be viewed as the horizontal sections of a bundle with respect to a flat connection.   The holonomy of a flat connection along a curve only depends on the homotopy class of the curve, hence gives a representation of the fundamental group.