In taking between 200 and 300 whitetails in my lifetime, I've tried just about every shot you can take. In my mind, there's no better place to shoot a deer than behind the shoulder.
Whether I'm shooting an arrow or a bullet, I'm trying to get both lungs. If I'm successful, the deer dies quickly and humanely, and it doesn't run far before going down. With a lung shot, there's usually a pretty good blood trail to follow.
If you shoot in the crease behind the deer's shoulder, you've centered the lungs. The best thing about the lung shot is the margin of error. You're looking at a vital area roughly the size of a 12-inch circle. If you miss low, you hit the heart. Miss a little forward and you hit the shoulder. Miss high and hit the spine.
Of course, not every shot is a perfect broadside shot. On a quartering-away shot, I try to aim so that my bullet or arrow strikes the offside shoulder. Otherwise, penetration into the body cavity might be too shallow. If the deer is quartering toward you, a rifle shot on the leading edge of the front shoulder will take out both lungs and possibly drop the deer in its tracks. Shooting at 3-D targets is a good way to become familiar with the best angle for quartering shots.
If the deer is facing you head on, you can take it with a rifle shot in the middle of the brisket, just below the base of the neck. To make this shot, however, you need a steady rest and a short distance to the deer.
Bowhunters must be more selective than gun hunters, but all hunters should know their limitations. I've passed up a lot of deer because I didn't feel good about the shot. When the deer is moving, or too far, or standing at the wrong angle, or I don't have a good rest, I'll pass up the shot.