What are natural questions to ask about an operad?

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 Dyer-Lashof 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 $Omega^nSigma^n X$ 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 Martin-Shnider-Stasheff 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 Dyer-Lashof 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.