This is a question I've asked myself a couple of times before, but its appearance on MO is somewhat motivated by this thread, and sigfpe's comment to Pete Clark's answer.
I've often heard it claimed that combinatorial species are wonderful and prove that category theory is also useful for combinatorics. I'd like to be talked out of my skepticism!
I haven't read Joyal's original 82-page paper on the subject, but browsing a couple of books hasn't helped me see what I'm missing. The Wikipedia page, which is surely an unfair gauge of the theory's depth and uses, reinforces my skepticism more than anything.
As a first step in my increasing appreciation of categorical ideas in fields familiar to me (logic may be next), I'd like to hear about some uses of combinatorial species to prove things in combinatorics.
I'm looking for examples where there is a clear advantage to their use. To someone whose mother tongue is not category theory, it is not helpful to just say that "combinatorial structures are functors, because permuting the elements of a set A gives a permutation of the partial orders on A". This is like expecting baseball analogies to increase a brazilian guy's understanding of soccer. In fact, if randomly asked on the street, I would sooner use combinatorial reasoning to understand finite categories than use categories of finite sets to understand combinatorics.
Added for clarification: In my (limited) reading of combinatorial species, there is quite a lot going on there that is combinatorial. The point of my question is to understand how the categorical part is helping.
Best Answer
Composition of species is closely related to the composition of symmetric collections of vector spaces ("S-modules"), which is a remarkable example of a monoidal category everyone who had ever encountered operads necessarily used. Applying ideas coming from this monoidal category interpretation has various consequences for combinatorics as well. For example, look at papers of Bruno Vallette on partition posets (here and here): I believe that already the description of the $S_n$ action on the top homology of the usual partition lattice was hard to explain from the combinatorics point of view - and for many other lattices would be impossible without the Koszul duality viewpoint.