The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra.
The distinction is rarely made in more recent writings.
Algebraic structures are defined primarily as sets with operations.
Structure preserving maps called homomorphisms are vital in the study of algebraic objects.
Advanced concepts: Representation theory