generally in differential cohesion
derived smooth geometry
The notion of étale morphism of schemes is the realization of the general notion of étale morphism for maps between schemes, hence it captures roughly the idea of a map of schemes which is a local homeomorphism/local diffeomorphism.
A central use of étale morphisms of schemes is that they appear as coverings in the Grothendieck topology of the étale site. The abelian sheaf cohomology with respect to these étale covers of schemes is accordingly called étale cohomology.
(A number of other equivalent definitions are listed at wikipedia.)
A jointly surjective collection of étale morphisms is called an étale cover.
Most of the pairs of conditions in def. 1 can be read as constraining the fiber of the morphism to be first suitably surjective/bundle-like (smooth, flat) and second suitably locally injective (unramified).
Specifically the first condition has an infinitesimal anlog: a formally étale morphism is a formally smooth and formally unramified morphism. These notions also have an interpretation in synthetic differential geometry and there they correspond to the statement that a local diffeomorphism is a submersion which is also an immersion of smooth manifolds.
A morphism is formally étale morphism if it is
(e.g. Milne, prop. 2.11)
A smooth morphism of schemes is étale iff there is an étale cover of the base scheme by open subschemes such that the pullback of the projection to each of them is an open local isomorphism of locally ringed spaces (and in particular, the pullback of the projection morphism is an étale map of the corresponding underlying topological spaces).
This disjointness picture of étale covers make them suitable for having nontrivial cohomology in situations where Zariski covers give vanishing cohomology.
This appears for instance as de Jong, prop. 3.1 i).
Such étale morphisms are classified by the classical Galois theory of field extensions.
This appears for instance as (Milne, prop. 2.1).
A proof is in (Deligne).
The latter is clear, since the very definition of
exhibits a finitely presented algebra over .
Now by the universal property of the localization, a homomorphism is a homomorphism which sends all elements in to invertible elements in . But no element in a nilpotent ideal can be invertible, Therefore the fiber product of the bottom and right map is the set of maps from to such that is taken to invertibles, which is indeed the top left set.
The classical references are
Lecture notes include
The local structure theorems are discussed in
and generally in E-∞ geometry in