I just discovered that something I've been working with has the structure of an operad. So I'm wondering what natural basic questions does one ask about operads? For example, if I knew I had the structure of a group, I'd ask if it is abelian or has torsion, etc. So what are these questions for operads?

There's a lot of things you could ask.
That'd be a start. 


Here are a few of my favorite questions about operads at the moment:



Hi Connie. Let me use your question as an excuse for an extended answer. A pair of brief papers "Definitions: operads, algebras and modules" and "Operads, algebras and modules", which are available at http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf and http://www.math.uchicago.edu/~may/PAPERS/handout.pdf (# 84,85 on my website) give several variants and reformulations of the original definition together with some history of antecedents, a variety of algebraic and topological examples, and the crucial relationship with monads that led me to coin the word "operad". There is also a discussion of the relationship to homological algebra, showing how the homological theory simplifies if you work over a field of characteristic zero and, in contrast, how operads encode homology operations (Steenrod operations and DyerLashof operation) if you work over a field of finite characteristic. Notes for a talk, http://www.math.uchicago.edu/~may/TALKS/SwitzerlandTalk.pdf, expand on the last point. The distinction of characteristic illustrates a general point. Operads are defined in any symmetric monoidal category, and the right questions to ask depend in large part on what category you are working in. It may make no sense at all to ask algebraic questions of a topological operad or topological questions of an algebraic operad. There is also a distinction to be made about questions to ask about operads and questions to ask about their algebras. Incidentally, groups are by design not examples of algebras over an operad: to define inverses, you need diagonals, and operads are not intended, or rather intended not, to incorporate such structure. The questions to ask also depend on what role your observation plays. Operads allow a taxonomy of certain types of algebraic structures, so the question may just be "what kind of structure am I looking at". But you might also want to ask whether the algebras you are looking at give simpler "approximations'' of more complicated or less accessible structures that occur "in nature". For example, spaces $\Omega^n\Sigma^n X$ occur in nature, but they can very usefully be approximated by the monads $C_nX$ associated to appropriate operads. You might also want to ask if operads can be used to define rigorously new structures that you want to study. A very recent example arose in work of Bertrand Guillou and myself in equivariant infinite loop space theory: there is an intuition of what a genuine strict symmetric monoidal $G$category should be, one that gives rise to a genuine $G$spectrum; the best definition we know is that such a category is an algebra over a particular operad in $Cat$ (see http://front.math.ucdavis.edu/1207.3459). Quite a few recent variants of the definition of an operad arose analogously. In algebra, very simple operads prescribe very natural and previously unstudied kinds of algebras. Loday and some of his students (I'm blanking on names) gave a number of examples. While one can ask questions about the homotopy theory of operads in general, using model category theory, that is perhaps my least favorite question to ask: it rarely cuts to the heart of the applications, excepting those in higher category theory, or so it seems to me. Model categories of algebras over particular operas do play a major role in many applications, albeit sometimes only implicitly. I'll stop here, since I could go on forever. One comment. While the MartinShniderStasheff book is a useful compendium, its treatments of different topics are not all at the same level, and you might well prefer less comprehensive treatments that better address your directions of interest. And people should be warned that the definition of an operad in that book is actually incorrect: it omits a crucial equivariance property that is of real importance in applications. For example, it plays a key role in the proof of the Adem relations for the Steenrod and DyerLashof operations. Benoit Fresse's book "Modules over operads and functors" gives a quite different take on operads, with a focus on modules over algebras over operads. 


I think this might be worthy of being a separate answer, so here I go. May has two versions of Geometry of Iterated Loop Spaces on his website, one type set and one in Tex. This is where operads were invented (even if in name only, I won't stress about history, and I mean no disrespect). It is helpful if only because it is an early treatment, do not read it too long, probably not past chapter 5. The diagrams will not seem natural until you have your two examples, the little ncubes operad and the endomorphism operad of a topological space. Then, if not before, read a little bit of Adams Infinite Loop Spaces, he uses props but thats ok. These are just little introductions stressing the homotopy theory side, there are other aspects, I just dont know them (I barely know the homotopy theory side). And when you get sad and stuck on something, just look up some new operad, here are two examples: the swiss cheese operad and the cactus operad. thanks Ryan for reminding me of May, almost everything he has published is legally available on his website, if not more! 


From a primarily combinatorial point of view of operads, here are a list of typical questions that you can ask about an operad $\mathcal{P}$.


