TITLE: Why are operads useful?
QUESTION [33 upvotes]: The question is not about where operads are used, I know that. It is about what makes them useful. For example, van Kampen diagrams are useful in combinatorial group theory because these are planar graphs and so one can use planar geometry (say, the Jordan lemma) to investigate the word problem in complicated groups. Similarly, asymptotic cones are useful in geometric group theory because they allow to study large scale properties of a discrete object (a group) by looking at small scale properties of a continuous object. I would like to know a similar answer for operads. 
 Update Many thanks to everybody for your answers. Unfortunately I can accept only one. So I just accept the first answer.

REPLY [7 votes]: There surely are many answers to this question... For me, one of the key reasons is that there are lots of situations where existing geometric and algebraic structures exhibit some kind of associativity. (Geometrically, think of gluing pants, like in Jeff's comment,  algebraically think of composing operations and/or cooperations.) The notion of an operad allows to formalise this observation, and treat objects like that as associative algebras in a certain monoidal category, - and since we know an awful lot of ways to approach usual associative algebras, this gives intuition of how to approach problems for these, more tricky objects. I think the analogy with your examples is quite clear.

REPLY [2 votes]: I'm not an expert by the way I could give you an answer based on my personal experience in my study of category theory.

Operads allow to make lots of constructions and to encode information of many mathematical object into an algebraic structure, this algebraic structure allows to work in a simpler way with the above mentioned objects.

In pratical I think operads are similar to homotopy/homology groups, which encode homotopical information of topological spaces and enable to distinguish such spaces in a simple way, studying algebraic structures. This is useful because is more simple classifying groups rather then topological spaces. 
More in general I think the usefulness of all such structures derive by the fact that usually (concrete) algebraic structures are easier to work with, but I emphasise that these are just my thoughts.