Algebra, which means "The union of parts" is a broad category of mathematics.
It's different from arithmitic because algebra generally contains an unknown.
For example, in the equation \(2x + 3 = 5\), the unknown is the commonly used variable x. (x is 1.)
Here is a list of categories of mathematics that has 'Algebra' in this. This is from Wikipedia, which has a much more extensive write-up on Algebra than anything you'll see here.
Elementary algebra, the part of algebra that is usually taught in elementary courses of mathematics.
Abstract algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated.
Linear algebra, in which the specific properties of linear equations, vector spaces and matrices are studied.
Boolean algebra, a branch of algebra abstracting the computation with the truth values false and true.
Commutative algebra, the study of commutative rings.
Computer algebra, the implementation of algebraic methods as algorithms and computer programs.
Homological algebra, the study of algebraic structures that are fundamental to study topological spaces.
Universal algebra, in which properties common to all algebraic structures are studied.
Algebraic number theory, in which the properties of numbers are studied from an algebraic point of view.
Algebraic geometry, a branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations.
Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions.
Relational algebra: a set of finitary relations that is closed under certain operators.
Just google Algebra. Not that hard.