Grothendieck topology

The notion of a Grothendieck topology or site is a category which has the features of open covers in topological spaces necessary for generalizing much of sheaf cohomology to sheaves on more general sites.

Examples

1. A standard topological space $X$ becomes a category $op(X)$ when you regard the open subsets of $X$ as objects, and morphisms are inclusions. An open covering of open subsets $U$ clearly verify the axioms above for coverings in a site. Notice that a presheaf of rings is just a contravariant functor from the category $op(X)$ into the category of rings.
2. The Small Étale Site Let $S$ be a scheme. Then the category of étale schemes over $S$ (i.e., $S$-schemes $X$ over $S$ whose structural morphisms are étale)