For a topological space $X$ a presheaf $F$ with values in a category $\mathcal{C}$ associates to each open set $U\subset X$ an object $F(U)$ of $\mathcal{C}$ and to each inclusion $U\subset V$ a morphism of $\mathcal{C}$ $\rho_{UV} : F(V)\to F(U)$ the restriction morphism. It is required that $\rho_{UU} = 1_{F(U)}$ and $\rho_{UW} = \rho_{UV}\circ \rho_{VW}$ for any $U\subset V\subset W$

A presheaf with values in the category of sets (or abelian groups) is called a presheaf of sets (or abelian groups). If no target category is specified, either the category of sets or abelian groups is most likely understood.

A more categorical way to state it is as follows. For $X$ form the category ${\bf Top}(X)$ whose objects are open sets of $X$ and whose morphisms are the inclusions. Then a presheaf is merely a contravariant functor ${\bf Top}(X)\to\mathcal{C}$