A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$ Should be used with the (group-theory) tag.

An abelian or commutative group is a group $(G,*)$ in which all elements commute: $$\forall a,b\in G\,\,, a*b=b*a\,.$$ Usually the product is denoted by $+$ in an abelian group, and the identity of the group by $0$.

Examples include the integers $\mathbb{Z}$ under addition, as well as the rationals $\mathbb{Q}$ under addition. In fact, every cyclic group is an abelian group. Non-examples include $S_3$, the symmetry group on three elements, as well as $\mathrm{SO}(3)$, the rotations in three dimensions.

The fundamental theorem of abelian groups says that all finite abelian groups are direct products of cyclic groups, themselves abelian.