A blood sport was any sport that involved the killing or injury of another, such as hunting.
In 2369, Benjamin Sisko defined it as "killing lower species for pleasure" and noted that centuries ago, Humans once practiced blood sports, and still did among some cultures. Unlike the Hunters with the Tosk, none of them would ever hunt a sentient lifeform. (DS9: "Captive Pursuit")
In 2374, The Doctor chastised Karr and the other Hirogen for causing a serious head injury to a member of USS Voyager's crew during their World War II simulations, despite their assurances to the contrary. He warned Karr that if he couldn't control his people, he ought to end their blood sport. (VOY: "The Killing Game")