Q:

What are some examples of interesting uses of the theory of combinatorial species? 

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. 

A:

Let's start from the beginning.   The main textbook on species is this one by Bergeron, Labelle,  and Leroux, all major experts in the field.   Even if you don't want to read the book, read the introduction by G.-C. Rota (which is interesting, enlightening and relatively short.   In there, Rota writes:

"I dare make a prediction on the future acceptance of this book.  At first, the old fogies will pretend the book does not exist.  This pretense will last until sufficiently many younger combinatorialists publish papers in which interesting problems are solved using the theory of species.  Eventually, a major problem will be solved in the language of species, and from that time on everyone will have to take notice." 

It has been 13 years since these words had been written (almost to the day), so it is perhaps time to revisit this prediction.   The "major problem" part clearly did not work out.   One can argue that it's too soon to judge.   Maybe.   Maybe not.   The first part, on "sufficiently many younger combinatorialists", is more interesting and probably arguable.   There are sufficiently many people using and referencing the book - it has over 300 citations on GoogleScholar.   And if you look at these citations, it becomes clear that the book has an extended influence over a large range of fields - the language and philosophy of species are clearly useful.   
On the other hand, in comparison with the "competition", the theory of species is clearly not doing so well.   Goulden & Jackson's "Combinatorial Enumeration" and Stanley's "Enumerative Combinatorics" have been cited about 1,000 times each (yes, both are older, but still).   If you go a bit further away from the field, Alon & Spencer's "Probabilistic Method" has been cited over 3,000 times...  
My conclusions: the answer to your question is both Yes and No.   The theory of species is clearly useful, but more like a good language to use, or a guiding principle of which roads to take and which to stay clear of.   When it comes to explicitly stated "practical" problems, people seem to prefer more directly applicable tools.   I would compare this phenomenon to the influence of complexity theory on enumerative combinatorics:  whatever you are enumerating, even if your problems are non-algorithmic and in a very classical combinatorial setting, it is still useful to know what is #P-completeness, simply because this gives you a different point of view on the objects you are enumerating, and sometimes it can also save you a bit of time by suggesting that the problem might be too general and thus have no explicit solution.   

A:

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.  

A:

Pietro, if you haven't done so by now, you really, really ought to read Joyal's paper. ( I can't understand why you would express skepticism before you'd even looked at the primary sources!)  
If there is a single application of species to be singled out from this wonderful article, it is Joyal's proof of Cayley's theorem. ( This proof was highlighted in Proofs from THE BOOK.)  But this is only one of the treasures in the paper that await those who take the trouble to read it.  
Much of the art of combinatorial thinking (at least in enumerative combinatorics) is knowing how to draw the correct pictures, and the theory of species can be seen as a step toward turning that art into a science, by formalizing directly the operations on structures which are implicitly coded by generating function techniques.  In different words, the basic functional operations on exponential generating functions are lifted to functorial operations on species.  At some level such insights must have been known to combinatorialists, but the theory of species serves to formalize them in the light of day, and no less a combinatorialist than Zeilberger has found species a significant source of inspiration. 