That should be read "B is etale over A". This happens when the map from A->B is an etale ring map, which means that its dual map is an etale morphism of affine schems from SpecB->SpecA, 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. We get flatness since the only stalk of specA 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->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-Git Chapter 7 section 85 (7.85) on page 366 .
http://www.math.columbia.edu/algebraic_geometry/stacks-git/
I'm sure it's also in Hartshorne.