The difference between an intension and an extension in contemporary logic comes from Frege, who distinguishes a concept's Sinn (sense) from its Bedeutung (reference).
The easiest way to illustrate the difference between sense and reference is through cases. The predicates "having a kidney" and "having a heart" clearly have different senses, but as a matter of contingent fact everything that has a kidney has a heart: hence we case the two terms are co-referential. The set of the renates = the set of the cordates, because set membership is extensional. First order logics, including set theory are all extensional in this sense. It doesn't matter whether we are talking about the set of the renates or the set of the cordates, because they are the same set, and we know they're the same set, because we define a set extensionally--i.e. purely by the reference of the terms. We don't need to know what the terms "renate" and "cordate" really mean, we just need to know which things they refer to.
An intensional logic, however we do need to know what the terms mean, because we want to ask not just about which things do, as a matter of contingent fact, happen to fall under those terms. For instance, modal logic is an intensional logic. Take the set of all things that have hearts. The sentence, "every renate is a cordate" can be checked by examining every item in the domain and checking that all the things that actually have kidneys also actually have hearts. But, "Necessarily, every renate is a cordate" can't be checked this way. Even if it is the case that all renates are actually cordates, this is just a contingent fact. So to establish the truth or falsity of the modal sentence requires me to have some kind of model that isn't just looking at the referents of the terms "renate" and "cordate", it's going to have to have something to do with knowing the meaning of the terms, etc.
It is because modal logic isn't extensional that lots of people in the first half of the 20th century were suspicious of it. (Think of Quine here.) That's the neat thing about Kripke style model theory for modal logic. It provides something that looks a lot like extensional semantic theory for an intensional logic. It doesn't make modal logic extensional, but it does open up the possibility of using nice model theoretical techniques that were designed for extensional logics on modal logic as well. Since getting access to those techniques, we can prove consistency, and some other metalogical results for some modal logics, which have led to renewed interest in these logics among philosophers.
There are other intensional logics as well. Modal logic is just one prominent example.