Originally it's a dog. The word is борзая. Breed of lean hunting dog which runs very fast. But it became slang (a bit more of criminal slang) which means someone acting not nice, or rude, or trying to break agreements, etc.... the reference to the dog is only partial and also it became mostly masculine борзой (a feminine form is also possible though). The dog is here Борзые собаки — Википедия