One of the historical motivations for sheaves have come from studying complex manifolds, complex analytic geometry, and scheme theory from algebraic geometry. This is because in all of the previous cases, we consider a topological space X together with a structure sheaf \mathcal{O} giving it the structure of a complex manifold, complex analytic space, or scheme. This perspective of equipping a topological space with a sheaf is essential to the theory of locally ringed spaces (see below).
