According to wikipedia, an algebraic structure is an arbitrary set with one or more finitary operations defined on it. From a model theory perspective, I understand this definition as: structure with no relations, only functions and constants (I added the constants to my definition, because I know that group, rings, have them). So for example an ordered field is not an algebraic structure (has relation), and a set isn't too (no function). Regardless of the exact definition, a massive part of mathematics is concerned with the study of these structures, and the relations between them. Geometry benefits from algebraic geometry, number theory from algebraic number theory, and so on. My question is, what do this sort of structures have in them, that makes us study and use them so much, and not other structures?
This is not a complete answer, but I'm not sure that your question can be answered. So here are some thoughts:
As Zhen Lin states in the comments, algebraic structures are somewhat simpler to deal with than structures with both functions and relations. Restricting attention even further, a class of algebraic structures is called a variety of algebras if it can be defined using only equations (like $x\cdot (y\cdot z) = (x\cdot y) \cdot z$). Examples of varieties are groups, rings, modules, boolean algebras... Varieties are the core topic of study in the field of universal algebra, and they are very nice classes for a variety of reasons (e.g. they are closed under product and coproduct, they have free objects...). One could answer that we spend so much time studying and using varieties of algebras simply because they have lots of structure which is interesting and useful.
But you should take the previous point with a grain of salt. First, one of the most important classes of algebraic structures, the class of fields, is not a variety of algebras, since the field axiom requiring inverses is not an equation: it has a more complicated logical structure. Second, one can just as well restrict attention to relational structures, structures with relations but no functions or constants. Relational structures are also very nice to deal with in certain contexts, and they are also quite important in mathematics (i.e. graph theory).
I don't think it's true that we spend a lot of time studying and using algebraic structures instead of other structures. I think we spend a lot of time studying certain algebraic structures, and also certain non-algebraic structures (like ordered fields or graphs), because they're particularly interesting and relevant. You'll find a lot more literature, for example, on graphs than you will on magmas. Magmas are perfectly good algebraic structures, but their definition is a bit to broad to be particularly interesting or useful (apologies to the magma theorists out there).
As for why the algebraic structures we study (e.g. groups, rings, fields, etc.) have turned out to be so fundamental, well, lots of people have written about this far more eloquently than I can. But I think the core reason is that they are abstractions of concrete things we care about in mathematics - groups capture the notion of symmetry, the integers are a ring, blah blah blah.