Barbarity is a vicious kind of cruelty. Deliberately inflicting terrible pain and suffering on other people is barbarity.
While true barbarity reaches the level of torture, this noun is also used as an exaggerated way to say "an absence of civility or culture." Feet on the table at dinner? That's utter barbarity! Leaving your trash on the ground after a picnic? Total barbarity! "Want of civilization" is actually the original definition of barbarity — "savage cruelty" came about a hundred years later, in the 1680s.