“Abelian” in this context means “commutative”. That is, if you have some operator * then it is abelian if a * b = b * a for all a and b in the domain of *. If that id not the case it is “non-abelian”.

The name “Abelian” is after Nils Henrik Abel, one of the founders of abstract algebra who died tragically young of tuberculosis brought on by extreme poverty.