Formally unramified

In algebraic geometry, a morphism ${\displaystyle f:X\to S}$ between schemes is said to be smooth if

(iii) means that each geometric fiber of f is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.

If S is the spectrum of an algebraically closed field and f is of finite type, then one recovers the definition of a nonsingular variety.

There are many equivalent definitions of a smooth morphism. Let ${\displaystyle f:X\to S}$ be locally of finite presentation. Then the following are equivalent.

A morphism of finite type is étale if and only if it is smooth and quasi-finite.

A smooth morphism is stable under base change and composition. A smooth morphism is locally of finite presentation.

A smooth morphism is universally locally acyclic.

Smooth morphisms are supposed to geometrically correspond to smooth submersions in differential geometry; that is, they are smooth locally trivial fibrations over some base space (by Ehresmann's theorem).

...
Wikipedia