An abelian group is a group where is commutative. In other words, the group operation satisfies the five axioms:
Closure: For all in , is defined and in . We abbreviate as .
Associativity: for all in .
Identity: There is an element such that for all in , .
Inverses: For each in is an element in such that .
Commutativity: For all in , .
The first four are the standard group axioms; the fifth is what distinguishes abelian groups from groups.
Commutativity gives us license to re-arrange chains of elements in formulas about commutative groups. For example, if in a commutative group with elements , we have the claim , we can shuffle the elements to get and reduce this to the claim . This would be invalid for a nonabelian group, because doesn’t necessarily equal in general.
Abelian groups are very well-behaved groups, and they are often much easier to deal with than their non-commutative counterparts. For example, every Subgroup of an abelian group is normal, and all finitely generated abelian groups are a direct product of cyclic groups (the structure theorem for finitely generated abelian groups).
I strongly recommend keeping to the standard term “abelian group,” even though “commutative group” would be more systematic and sensible. The term “abelian group” is universal—I don’t know a single mathematician, book, or paper that uses the term “commutative group”—and people comparing what they read here to what they read anywhere else are just going to be confused, and/or are going to confuse third parties when they ask questions.