An Algebraic Structure is a set, a collection of operations over that set and a list of axioms.
The operations are usually binary.
A more precise definition included that the operations were of finite arity.
Arity is the number of arguments in a function. A binary operation has an arity of 2.
Modern Algebra examines the algebra of various mathematical structures with the goal of gaining a broader and more sophisticated understanding of algebraic structures. Topics include Groups, Rings, Fields and other main algebraic structures.