Reading the comments, I just realized that part of your question asks about what sets of intermediate cardinality would look like. As Richard and John Goodrick pointed out, there are straightforward definitions of ordinals of intermediate cardinality. But I assume the question is more about whether there are any sets of real numbers with intermediate cardinality, and what such sets of reals would look like.
It's not too hard to use the ordinals to cook up some sets of real numbers. (For instance, there's probably a definable way to represent each countable ordinal by a real number, and then the set of representations will itself be of intermediate cardinality.) But as far as I know, no one has come up with any direct characterization of a set of real numbers that could have intermediate cardinality.
There are a lot of negative results though. The first is the Cantor-Bendixson theorem, which states that no closed set can have intermediate cardinality. (The theorem shows that every closed set is either countable or has a http://en.wikipedia.org/wiki/Perfect_set_property>perfect subset. Since perfect sets have cardinality of the continuum this means they can't be intermediate.)
In fact, some difficult work of Martin and others in the '70s showed (using just ZF) that Borel determinacy is true, which among other things entails that every Borel set is either countable or has a perfect subset. If we further assume projective determinacy (which set theorists tend to believe) then the same is true for projective sets.
Thus, any set of intermediate cardinality has to be pretty weird. It can't be closed, it can't even be Borel, and (if set theorists are right about projective determinacy) it can't even be projective. Thus, it must be pretty crazy, just as we know about non-measurable sets. (Though there's no guarantee that intermediate cardinality goes along with non-measurability - Martin's Axiom guarantees that in fact every set of intermediate cardinality is measurable and has measure 0.)
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