Typically, an arrow kills by cutting blood vessels (arteries and veins), resulting in massive blood loss. Blood pressure drops, cutting off oxygen to the brain and causing death. This is called hemorrhagic shock.
An animal must lose about 1/3 of its blood to die of hemorrhagic shock, but that can vary depending on how fast the blood is lost. There is approximately one ounce of blood per pound of body weight in the circulatory system of animals like deer.
The time it takes for an animal to die from bleeding can vary from a matter of seconds to several hours, depending on the animal's size, how many vessels are cut, and whether they are arteries or veins.
Arrows also can kill by puncturing the lungs. If both lungs collapse, the flow of oxygenated blood to the brain is halted, the animal becomes unconscious within seconds, and it dies before it can bleed to death.
Arrows can kill by disrupting the heart muscle, which instantly stops the flow of blood to the brain.