A barbarity is a very cruel act — one piece of cruelty you can point to, such as the barbarities of a war. It can also mean that cruel quality in general, but learners meet it most as 'a cruel thing that was done.' It is a formal, serious noun; you will find it in news and history writing more than in chat. It sits next to barbaric: the act is a barbarity; the act is barbaric. Though actually, because it sounds so close to barbaric, students often try to use it as an adjective. It is not one. You can say a barbaric attack, or the barbarity of the attack, but not 'a barbaric' when you mean the act.