There are a few ways that getting shot can kill/incapacitate a person. 1: a person gets shot, the wound gets infected, and the person dies. That takes awhile.
2: blood loss causes the person to loose consciousness. This usually takes awhile.
3: the round causes skeletal damage that makes the person unable to run/fight. Some examples would be breaking a leg, hip, or arm.
4: the round takes out the brain.
5: the round damages the heart/lungs. This does not necessarily stop the person instantly. Even if the heart is stopped the person may still have enough oxygen to keep going for 15 or 20 seconds, which may be more than enough time for them to shoot/stab/hit you.