For sufficiently nice topological spaces $X$ (e.g., locally connected for the last two categories to be equivalent, and semilocally simply connected and locally path-connected for all three to be equivalent), the following three categories are equivalent:
Functors from the fundamental groupoid of $X$ to the category of sets;
Covering spaces over $X$;
Locally constant sheaves of sets on $X$.
This is an extremely primitive baby version of the Riemann–Hilbert correspondence.
References specifically for this elementary case are sparse, but there is an extensive discussion on locally constant sheaves at the nCafé.