Sheaf

Definition
A sheaf is a presheaf that satisfies both of the following two additional axioms:

(Locality) Suppose $U$ is an open set, ${U_{i}}_{i \in I}$ is an open cover of U $U$ , and $s, t \in F (U)$ are sections. If $s |_{U_{i}} = t |_{U_{i}}$ for all $i \in I$ , then $s = t$ .
(Gluing) Suppose $U$ is an open set, ${U_{i}}_{i \in I}$ is an open cover of $U$ , and ${s_{i} \in F (U_{i})}$ is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if $s_{i} |_{U_{i j}} = s_{j} |_{U_{i j}}$ for all $i, j \in I$ , then there exists a section $s \in F (U)$ such that $s |_{U_{i}} = s_{i}$ for all $i \in I$ .
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They can be locally free sheaf.
Related: localization of a ring.

Reflection: from here.
Recall the primary motivation of, say, Algebraic Geometry: a geometric space is determined by its algebra of functions. Well, actually, this isn't quite true --- a complex manifold, for example, tends to have very few entire functions (any bounded entire function on C is constant, and so there are no non-constant entire functions on a torus, say), so in algebraic geometry, they use "sheaves", which are a way of talking about local functions. In real geometry, though (e.g. topology, or differential geometry), there are partition of unity, and it is more-or-less true that a space is determined by its algebra of total functions...
(From a physics point of view, it should be taken as a definition of "space" that it depends only on its algebra of functions. These functions are the allowed "observables" or "measurements" --- if you can't measure the difference between two systems, you have no right to treat them as different.)