To understand this definition, and what it has to do with the more familiar definition of a presheaf, let $T$ be a site. Then a presheaf maps the “open sets” to objects in $C$ and provides restriction maps whenever the underlying “open sets” have an “inclusion”. Of course, on a site, there may be many non-isomorphic inclusions.

In practice, to avoid set-theoretic difficulties, one usually fixes a universe $\mathcal{U}$ and then requires $T$ to be a small category.

Let $T$ be a site, and suppose either $C$ is an abelian category or $C$ is the category of sets.

This “exactness” really needs explaining. First of all, for every $i$, we have a morphism $F(U)\to F(U_{i})$ obtained from the covering. The map on the left (call it $f$) is the product of all these maps. Second, for each pair $i$ and $j$, we have a morphism $g_{1}\colon F(U_{i})\to F(U_{i}\times_{U}U_{j})$ and a morphism $g_{2}\colon F(U_{j})\to F(U_{i}\times_{U}U_{j})$ obtained from the two projection maps of the fibred product. The two maps on the right-hand side are obtained as products of all the $g_{1}$ and $g_{2}$ respectively.

If $C$ is the category of sets, then the exactness condition requires that the left-hand map is injective and that the two right-hand maps agree exactly on its image.

If $C$ is an abelian category, then the kernel of the left-hand map should be zero, and the kernel of the difference of the two right-hand maps should be its image. Perhaps this would be clearer if the diagram added a “$0\to$” on the left, but this usage is very standard.

This definition of a sheaf is the motivation for the definition of a site: a site captures precisely those features of a topological space that are required to form sheaves. The category of sheaves on a topological space is an example of a topos; the original definition of a topos was “a category equivalent to the category of sheaves on a site”, although the concept was generalized in later work. The motivation was that while a site may be badly behaved and pose set-theoretic difficulties, the topos associated to it is generally better behaved.

The category of sheaves on a site has a long list of nice properties; among them is “has enough injectives”, so that derived functors can be extracted and cohomology calculated. This is how, for example, étale cohomology is obtained.