Here is my own attempt at organizing the interrelationships for Groups and Rings. But as I said, I am an amateur in the subject at best, and so this may be incorrect in some places. If so, I would be really grateful to any corrections anyone can give me.
The way I've chosen to organize the structures is as a downwardly flowing web, whose nodes are the various algebraic structures, and whose directional edges denote the axioms that need to be added to the upstream structure in order to produce the downstream structure. This is done in a symmetric way, so that the axiom together with its upstream structure is both necessary and sufficient for producing the next downstream structure.
Groups
Rings
Note: I know there are different methods to define Rings (e.g. whether it includes a multiplicative identity or not). In the following graphic I adopt the convention where a ring does contain $1$, and those that don't are called "rngs".
Also, because some of the ring-like structures (e.g. near-rings) branch out early in the hierarchy before reaching an "officially" named structure like semirings, I had to invent some of my own names for various primitive ringoids to provide the nodes for the branching. Such homemade names I have enclosed in "quotes" to distinguish them from the more standard names.
These graphics were generated using draw.io
Informally speaking, an abelian group is a place where you can sum. It's the essence of what defines the sum of ordinary 'counting numbers'.
This allows you - after some exposure - to handle very abstract objects with the same familiarity you have for the latter.
As for any abstract definition, it is not going to make sense unless you have some examples at hand. And as for most abstract definitions, the examples came first. Many, many objects come with a binary operation that is associative and commutative (with unit and inverses) so we gave it at name.
Any "number system" $R$ (technically, I mean a ring here) has a notion of sum $+$, and $(R,+)$ is an abelian group. This includes very familiar number systems such as the integers, rational, real and complex numbers.
But is also includes for example matrices over these number systems. In general, product of matrices is known to depend on the order of the factors, but not their sum. Hence, we can sum matrices 'as if they were numbers'.
Another example is the "number system" $\mathbb{Z}_{12}$, which behaves like hours in a clock. You can think of it as the numbers from $1$ to $12$, but here e.g. $11+4 = 3$. It may be uncomfortable to work with this at first, but knowing that $+$ behaves similarly to ordinary numbers helps.
The list goes on, of course, but it becomes more abstract.
Another thing which may be useful to think about is... well, non-abelian groups. As we said before, in general for matrices $A,B$ we have $AB \neq BA$. Things get non-abelian really quickly in real life too: it is not the same to put your jacket on first and then your t-shirt than doing so in the reverse order.
Commutativity is far from a 'given', hence it is important to know when it does hold. It makes some things easier to organize. But as I said before, if you are not convinced that groups are important in the first place then it may not be clear why them being abelian is a thing to care about.
Best Answer
There absolutely are non-associative structures! Relevant terms here include "loop" and "Lie algebra." For concrete examples, consider octonion multiplication or the "midpoint" operation on points in $\mathbb{R}^n$ (note that the latter is commutative but not associative!). There are also weakenings of the associative law, like power associativity or alternativity. The study of nonassociative structures is very much a thing.
However, there is definitely an important sense in which associativity is special, namely that it lines up with function composition (note that function composition is not commutative: $(x+1)^2\not=x^2+1$ in general). This is important since we generally run into groups not "in a vacuum" but rather as a component of a richer object - namely a group action. A group action is basically a way of representing elements of a group $G$ as "structure-preserving" maps on some other structure $A$. The point is that this is generally the way groups emerge in the first place: consider for example Galois theory, where we're not interested in $Gal(K/F)$ on its own so much as we're interested in the action of $Gal(K/F)$ on $K$.
And the other direction is important too: given a group $G$ we can often gain a better understanding of $G$ by thinking about its possible actions. An early example of this is Cayley's theorem, where we look at a simple action of a group on itself, and representation theory is built around the idea that we can learn a lot about a group by looking at the various ways it can act on vector spaces.
The idea of an action - where elements of the acting structure represent functions on the acted-on structure, and the operation of the acting structure represents composition - automatically enforces associativity since function composition is associative. Conversely, if we try to whip up a notion of "action" which does not automatically enforce associativity - that is, a notion of a non-associative magma acting on a structure $X$ in such a way that $[a*(b*c)]x\not=[(a*b)*c]x$ in general - things get messy. This doesn't make non-associative structures uninteresting, but it does mean that the natural idea of "action" is really only appropriate to associative structures, and this does wind up being a big deal.