Is a group always abelian?

Could you please elaborate on whether a group is inherently abelian or if there are certain conditions under which it can be considered so? It's been intriguing to ponder whether the commutative property of group elements, where the order of their operation doesn't affect the result, applies universally or if there are exceptions to this rule. Could you provide insights into the circumstances where a group might not be abelian, and what characteristics distinguish abelian groups from non-abelian ones?