I am a second-year graduate student of pure mathematics. Like most graduate students, I have been exposed to many types of algebraic structures, and it seems standard for the major emphasis, if not all the emphasis, to be on groups, rings, R-modules, and categories. These are very rich structures with very interesting properties, but, when I zoom out to see the big picture of algebraic structures, I have always wondered why a particular set of defining properties make for such a rich structure, while another set of properties gives less "interesting" structures, or nothing worth teaching at all.
For a motivating example, at least one operation over a set (or class, whatever) that is closed under the operation definitely seems necessary to talk about anything meaningful; however, why is the particular combination of 1.having inverse elements, 2.having an identity element, and 3.associativity so much more rich (a group) than simply replacing associativity with commutativity (a structure I don't even know a name for)? In fact, a good question I have always had is why associativity is so much more prevalent and necessary than commutativity? As another motivating example, we are taught 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 about?
Finally, along the same path, I predict many answers will mention how the definitions for certain objects are better stated in category-theoretic terms (a group is a groupoid with one object); however, this simply moves the problem, because 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? Indeed, categories are extremely powerful objects, I guess just don't know why the particular combination of its defining properties is much more "powerful", deep, and pervasive than another combination of properties.