If f is a map of local rings $$f:A\rightarrow B$$ is étale iff it is flat and unramified (check out Bhargav Bhatt's notes at the stacks project link text). If A is a field and B is finite over A, then f is étale iff B is isomorphic to a finite product of separable field extensions of A (see proposition I.3.1 of Milne's book "Étale cohomology"). More generally, for f any ring homomorphism, check out definition II.1.1 of SGA 4.5 (B is a finitely presented A-algebra and satisfies a Jacobian criterion is a possible definition. Or B is a finitely presented A-algebra and B is flat and the relative differentials are trivial). The definition comes down to "smooth of relative dimension 0".

If $f$ is a map of local rings $$ f\colon A\rightarrow B $$ is étale iff it is flat and unramified (check out Bhargav Bhatt's notes link text). If $A$ is a field and $B$ is finite over $A$, then $f$ is étale iff $B$ is isomorphic to a finite product of separable field extensions of $A$ (see proposition I.3.1 of Milne's book Étale cohomology). More generally, for $f$ any ring homomorphism, check out definition II.1.1 of SGA 4.5 ($B$ is a finitely presented $A$-algebra and satisfies a Jacobian criterion is a possible definition. Or $B$ is a finitely presented $A$-algebra and $B$ is flat and the relative differentials are trivial). The definition comes down to "smooth of relative dimension 0".

$$f:A\rightarrow B$$ is étale iff it is flat and unramified (check out Bhargav Bhatt's notes at the stacks project link text). If A is a field and B is finite over A, then f is étale iff B is isomorphic to a finite product of separable field extensions of A (see proposition I.3.1 of Milne's book "Étale cohomology").

If f is a map of local rings $$f:A\rightarrow B$$ is étale iff it is flat and unramified (check out Bhargav Bhatt's notes at the stacks project link text). If A is a field and B is finite over A, then f is étale iff B is isomorphic to a finite product of separable field extensions of A (see proposition I.3.1 of Milne's book "Étale cohomology"). More generally, for f any ring homomorphism, check out definition II.1.1 of SGA 4.5 (B is a finitely presented A-algebra and satisfies a Jacobian criterion is a possible definition. Or B is a finitely presented A-algebra and B is flat and the relative differentials are trivial). The definition comes down to "smooth of relative dimension 0".