The term "game" in dog temperament refers to a dog that seemingly has a disregard for injury, exhaustion, and death when in an aggressive state, particularly when hunting and fighting. The term has historically been used to describe the temperament of fighting dogs that have derived from the old bulldog of England that was used to bait bulls for sport.
It was common practice to dismember these bulldogs by cutting off a front limb, as they remained clamped onto the faces of an exhausted and defeated bull. Bulldogs that would endure this practice and remain in a fighting state became valued breeders.
Although this practice ended with the outlawed sport of bull baiting this trait was passed along and tested in other ways, such as fighting rings against other dogs, during the development of the pit bull style dogs that derived from these early bull dogs. According to formal rules of most fighting rings, which were legal for some time, it was not the most domineering dog that won a fight, it was the dog that stopped fighting that would lose. So it was possible and common for a very injured dog with broken limbs to win a fight because it didn't stop fighting a "stronger" dog that eventually gave up due to exhaustion.
The trait was also interestingly passed along to the greyhound who were crossbred with bulldogs to make them more persistent on their rabbit hunts. It was reported that some greyhounds would run themselves to death if they didn't catch the rabbit after this cross took place.
It is important to note that NO breed of dog is automatically game just because the trait was historically encouraged through breeding. The term "game bred" is used when a breeder is making an effort to maintain the trait through carefully breeding "game" parents.
These two videos show some reality of "game bred" out of the mouths of the type of people that have bred for gameness for centuries: