This entry is about étale morphisms in the sense of algebraic geometry. The word étale map is preffered in differentiable and topological contexts, see étalé space for the topological version.
Recall that for a smooth morphism of schemes we can define a relative dimension in every point . A morphism of schemes is étale if it is a smooth morphism and if it is of relative dimension . Equivalently, it is flat and unramified. The property of being étale is preserved under pullbacks along any morphism of schemes; a composite of étale maps is étale.
There is a weaker notion of a formally étale morphism. A morphism is formally étale if it is formally smooth (satisfying an infinitesimal lifting property) and formally unramified. These are sheaf-like properties, what can be formalized in the language of Q-categories (monopresheaf and epipresheaf properties on the -category of nilpotent thickenings).
Étale morphisms are used to define small and big étale sites? and étale cohomology. Étale topology has similar cohomological properties to complex analytic topology?, and in particular it is much finer for cohomological purposes than the Zariski topology?.
A smooth map 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.