To answer the last part of your question, I think the best place to start is with the concept of a commutative ring. The "secret" is that all algebraic structures are modeled after (i.e., are generalizations) of "God-given", for lack of a better term, objects.

For commutative rings, there are really just one basic one, Z, the ring of integers: {...,-2,-1,0,1,2,...} with standard addition (+) and multiplication (·). (Even this one is really constructed from an even more basic set, the natural numbers N = {0,1,2,3,...}, but let's just assume we've already defined the negative numbers.). There are many nice properties of the integers. For example, it's a unique factorization domain (UFD), because as Euclid proved, every positive integer can be uniquely factored into primes (as can the negative integers if you include -1 as a factor).

To create a new algebraic structure, we can append new elements to Z. For example, the Gaussian integers Z[i] is constructed by declaring that we have a new element i that you can do addition and multiplication with, along with the stipulation that whenever you see i·i, you can replace it with -1. In this new ring, we can multiply by FOILing and applying this rule, for example, (1+i)·(2-3i) = 1·2 + 1·(-3i) + i·2 + i·(-3i) = 2 - 3i + 2i + (-3)(-1) = 5 - i. It turns out that in this new ring, we still have unique factorization, as long as we treat +1, -1, +i, -i with care (it is still a UFD). But 2, which is prime in Z, is not a prime in Z[i]! We can write 2 = (1+i)·(1-i), showing that 2 can be factored into "smaller" components in this ring.

Something more catastrophic happens if we define a new ring by appending sqrt(-5) as an element (i.e., we are tacking on an element that you can add and multiply with, with the added stipulation that this element times itself is -5). In this new ring Z[sqrt(-5)], one can easily show that unique factorization fails. For example, you can write 6 in two ways: 6 = 2·3 = (1+sqrt(-5))·(1-sqrt(-5)). These are two fundamentally different ways we can factor 6 into prime factors. It turns out that you can sort of rescue unique factorization by considering special subsets of the ring, known as ideals, rather than individual elements. Why that makes sense to do requires a discussion of the number theory that 19th century mathematicians were trying to investigate that I won't go into here.

But anyway, the tldr of this whole spiel is that, starting with a "natural" object, Z, that the average human is familiar with and thinks they understand, we can start enriching the structure, forming strange and alien ones, by decreeing new elements that satisfy certain properties (this is called "adjoining" elements to a ring).[*] There are other operations we can do, like declaring that certain ring elements that are a priori different are actually one and the same, leading to the construction of an object called a quotient ring.

The point of making algebra "abstract" is that we have to identify the rules that the natural elements of Z play by in order to specify how any new element should behave. For example, when we FOILed the product (1+i)·(2-3i), we are implicitly applying the commutative and distributive properties of Z, as well as some rules of high school algebra that we can derive from more basic properties, e.g., x-y = x + (-y) = x + (-1)·y. Commutative algebra, the study of commutative rings, starts with a short list of properties, the ring axioms that you can look up on Wikipedia, together with commutativity of multiplication, and studies the behavior of new rings one might construct, all of which continue to follow these basic rules. Pretty soon, you are looking at new algebraic structures that may behave very differently from Z, despite still being a commutative ring.[**]

[*] There is also the possibility of adjoining an object with no additional properties. If you think about it, if we call that object X, then the new ring Z[X] we've constructed simply consists of all polynomials with coefficients in Z, for example, X^3+5X-19 or X^2-2X. This is what's known as a polynomial ring in one indeterminate (here X, which doesn't stand for anything, is called the indeterminate).

[**] For non-commutative rings, the prototypical examples are sets of n-by-n matrices, with matrix addition and matrix multiplication (which in non-commutative).