$\DeclareMathOperator{\Spec}{Spec}$ That should be read "$B$ is etale over $A$". This happens when the map from $A\to B$ is an etale ring map, which means that its dual map is an etale morphism of affine schems from $\Spec B \to \Spec A$, which is defined:
http://en.wikipedia.org/wiki/Etale_morphism
As with most things in ring theory, this condition is somewhat more trivial when $A$ is a field $k$. We get flatness since the only stalk of $\Spec A$ is $A$ ($\Spec A$ has one point), which is a field, so all of its modules are free, and hence flat. Unramifiedness will not always hold, but it's also lot easier because $k$ is a field. If $k$ is of characteristic zero, the extension is automatically separable, so then we only need to restrict to it being finite.
There's a more direct definition which says that the morphism $A\to B$ is a smooth ring map with relative dimension zero.
If you'd like to read a section on them in more generality, you can check out Stacks Project Tag 00U0.
I'm sure it's also in Hartshorne.
