What is Abstract Algebra essentially?

We learn math with numbers early on. We learn how to apply operations to numbers to get new numbers. We learn rules, and consequences of those rules. All of that is pretty straightforward.

But, the real numbers are not the only things we might want to examine in detail. The properties of how elements interact under operations is a more general, abstract notion of what we do with numbers when we do algebra.

For instance, maybe we want to examine what a shape looks like if we rotate it around. Maybe you run a supply chain, and you need to build 4 widgets, but only some of those widgets need to be built in a certain order. Could you re-arrange things to make it more efficient? Maybe we want to explore structures that have a fundamental periodicity, like the time of day.

Over time, we have constructed concepts of structures that elements can belong to, and notions of operations on these structures. These structures -- groups, fields, rings, monoids, modules, vector spaces, etc. -- don't have a natural set of rules, per se. We make up those rules (aka axioms), but we have found that many natural concepts adhere to those rules.

This is all well and good but somewhat useless until you learn about isomorphism. Exploring what a group is or what a ring is is fine. But the richness of abstract algebra comes from the idea that you can use abstractions of a concept that are easy to understand to explain more complex behavior! Adding hours on a clock is like working in a cyclic group, for instance. Or manufacturing processes might be shown to be isomorphic to products of permutations of a finite group.

Abstract algebra is what happens when we want to explore consequences of rules and properties on collections of objects of any type -- hence the term "abstract!"

In "concrete" algebra one works with things like integers, rational numbers, real numbers, complex numbers, matrices, quaternions, permutations, polynomials, geometric transformations (e.g. isometries, similarities, reflections, inversions, projectivities, etc.), etc., subject to operations like addition, multiplication, and composition.

In "abstract" algebra one says "suppose we have a set of objects (which could be numbers, matrices, permutations, geometric transformations, etc., but we will not say what they are) and certain operations (which could be addition, multiplication, composition, etc., but again in certain contexts we won't say what they are) that are assumed subject to certain algebraic rules, such as commutativity, associativity, distributivity, the existence or non-existence of identity elements and inverses, closure or lack of closure, etc.). Then one deduces consequences of those algebraic laws. Statements that say that something is always true are deduced from algebraic laws without considering the concrete nature of either the objects or the operations. Statements that say that something is not always true are often deduced from concrete examples, involving numbers, matrices, polynomials, permutations, etc.

In abstract algebra, the examples are concrete, but the derivations of general results come from the rules of algebra without the concrete nature of the operations or the things they operate on.

Pinter in "A book of abstract algebra" says it thus:

Thus, we are led to the modern notion of algebraic structure. An algebraic structure is understood to be an arbitrary set, with one or more operations defined on it. And algebra, then, is defined to be the study of algebraic structures. It is important that we be awakened to the full generality of the notion of algebraic structure. We must make an effort to discard all our preconceived notions of what an algebra is, and look at this new notion of algebraic structure in its naked simplicity. Any set, with a rule (or rules) for combining its elements, is already an algebraic structure. There does not need to be any connection with known mathematics. For example, consider the set of all colors (pure colors as well as color combinations), and the operation of mixing any two colors to produce a new color. This may be conceived as an algebraic structure. It obeys certain rules, such as the commutative law (mixing red and blue is the same as mixing blue and red). In a similar vein, consider the set of all musical sounds with the operation of combining any two sounds to produce a new (harmonious or disharmonious) combination.