TITLE: Why are particular combinations of algebraic properties "richer" than others?
QUESTION [23 upvotes]: Pedagogically, when students are exposed to algebraic structures it seems standard for the major emphasis, if not all the emphasis, to be on groups, rings, R-modules, and categories. These are rich structures with interesting properties, but in the big picture, I have wondered why some defining properties make for a rich structure, while other properties gives less interesting structures, or nothing worth teaching at all.
As a motivating example, a set (or class, whatever) that is closed under some operation seems necessary to talk about anything meaningful; however, why is the particular combination of 

Having inverse elements
Having an identity element
Associativity

more rich (a group) than simply replacing associativity with commutativity (a structure I don't even know a name for)? I have also wondered why associativity is much more prevalent than commutativity. As another motivating example, we teach much about groups and rings but why not loops, monoids, semilattices, and near-rings? What makes the former set either richer in structure or more pedagogically sound to teach?
Even in category theory I can ask what makes the specific combination of defining properties of a category so great. —why associativity and not commutativity? —why categories and not semi categories? I wonder why its particular combination of defining properties is more "powerful", deep, and pervasive than another combination of properties.

REPLY [4 votes]: Since you already got 3 deep and pervasive answers, I want to concentrate on a single minor point.

I can ask what makes the specific combination of defining properties of a category so great?

In my humble opinion, almost nothing: you will certainly be interested in the work a friend of mine and I (in minor extent) are laying down where we study multi-object partial magmas and recover a great deal of classical "category" theory: you get these things called plots where your composition is not defined for each pair of consecutive arrows, and when it's defined it is possibly non-associative. Finally, you don't have identities everywhere.
"You fool, nothing good can come out of these poorly behaved, thorny things!"
:) not at all. 
You can define "isomorphisms" (yes, without identities), and notice that "being an isomorphism" and "admitting an inverse" are different notions in this world, and that they collapse in the categorical world (a category is an associative plot, where the composition is defined and every object has a 1, in the same vein a monoid is an extremely smooth partial magma). You can then define isoids, i.e. plots where every arrow is an isomorphism.
We're even able to define morphisms of plots (p-unctors), natural transformations (trimmings, if I remember well the name Salvatore and I chose), adjoints, limits, and a chain of free-forgetful adjunctions which connects the category (it is a category) of plots to the category of associative plots, semicategories [in this case we've even two different adjunctions for two different fully faithful embeddings], and categories. Other things in the to-do list: what's a $n$-plot? How can one define localization of a plot with respect to a family of arrows? How about simplicial stuff, how about enrichments (whatever this means)?
"You are only children playing with symbols! Examples, Examples!!"
:) Functional analysis and symplectic geometry provide "natural factories" of examples of such structures. For example, one of our two unitization functors applied to the category of symplectic relations gives precisely the Woodward-Wehrheim category.