The taste of "game" (venison is the stereotype) is different from the taste of "domestic" meats such as beef, pork or poultry. The expression "gamy" describes meat that tastes or smells different, perhaps stronger, than the bland, familiar tastes. "Gamy" is often extended to mutton (sheep) or lamb, goat, camel, and animals such as mountain sheep, moose or bear, that people do not ordinarily eat. Range-fed (grass-fed) beef tastes different from the meat of feed-lot cattle, and we say that the range-fed beef is "gamier."
The taste of a particular type of meat is determined by: the characteristic chemistry and type of muscle and fat, of the breed of animal (and others of its family), the animal's diet (grasses, leafy plants, flowers, fruit, bark, etc.) and nutrition (protein, minerals.) The same effect of diet is even more obvious in the milk of nursing females; and the animal's movement and activity patterns. Range fed cattle get more exercise than do animals kept in feed-lots, so the leg and rump meat of grazing cows has more myoglobin and is "gamier." This is why poultry legs, thighs, wings and backs are darker and have stronger flavor than the breasts.