Abstract algebra studies sets equipped with operations and the laws those operations satisfy. Where elementary algebra manipulates numbers and symbols, abstract algebra asks: What features of addition, multiplication, or symmetry are essential—and what new worlds open if we keep the laws but change the underlying set? The answer yields a unifying language for mathematics and a toolkit used across physics and engineering.
The core idea is to take a set 
and equip it with one or more operations (e.g., +, ⋅, inverse, scalar multiplication). These operations may posses properties such as associativity, commutativity, distributivity and identities. This allows the study equations and structure-preserving maps between algebras (homomorphisms, isomorphisms).
These algebraic structures are essential to solve equations and study linear systems such as vector spaces which are heavily used in finite element models, quantum mechanics and special and general relativity to name a few. They provide deep explanations about every day processes when using numbers.
Since operations are bestowed on sets, set theory is essential to the understanding of algebra.