Reposting my answer to a similar question, but I think it gives a thorough explanation of how someone can be instantaneously killed.
Three ways to kill someone (which are certainly interrelated): take out their nervous system, take out their circulation, or take out their breathing.
Quickest way to go from alive and conscious to dead and done is to sever the brainstem, which acts as Grand Central Station for all nerve activity in the body – consciousness, breathing, heart rate, blood pressure, non-reflexive movements. The brainstem lies inside the skull, just above the junction with the neck. Feel that bony protrusion just behind your earlobe? Go from one side to the other, and you’ll cross right through the brainstem. As you can imagine, not an easy spot to get to if you want to kill someone. I’ve read a few novels where someone has put a hunting knife right through the base of the skull to kill somone. I suppose it would work, but you’d have to have the person’s neck flexed (chin to the chest) and still be very, very strong and accurate to get to that point. More commonly, this is done with a bullet. Even then, a small cal pistol or rifle has to be aimed pretty darn well in order to make this an instantaneous death. Precision is slightly less important if using special ammunition (e.g. hollow point bullets) or very powerful weapons (e.g. sniper rifles), where enough tissue is damaged as the bullet passes through the skull to injure the brainstem. Any of these methods would destroy the brainstem quickly enough that death would be essentially instantaneous.
A properly executed hanging breaks the neck just below the brainstem. This in turn causes rapid swelling of the spinal cord and disrupts blood flow and nerve signals in the brainstem, leading to death. May take a few seconds, but still considered a very swift death. However, hanging has gone out of favor as a form of capital punishment. Too short or too slow a drop and the victim dies by strangulation/suffocation, which can take 20 minutes or longer. Too fast or too long a drop and the victim gets decapitated. Does the job, but messy and unpleasant, even if you support capital punishment.
What’s another way to take out the brainstem? Kill the cells by depriving them of oxygen and saturating them with carbon dioxide. This is where circulation and breathing come in.
Breathing is a quicker discussion: lungs help trade oxygen for CO2 in the blood. Since a killer/executioner can’t stop that process directly, they instead would have to cut off the airway. Asphyxiation by hanging was just mentioned. Other methods would be similarly slow – choking, pillow over the face, etc. You could punch holes in the rib cage next to the lungs to disrupt the mechanics of breathing (this was done in Dan Brown’s Angels and Demons – look up pneumothorax for the exact mechanics), but again it would be a slower death.
Now, for circulation. Blood gets pumped by the heart, so if you deprive the body of the blood or the pump, you won’t nourish and cleanse the brainstem tissues and the brainstem will die. A stab wound to the heart won’t directly cause it to stop beating. Instead, blood will leak through the puncture wound and fill the pericardium (a thin layer of tissue that surrounds the heart, but is separate from it). As blood fills the pericardium, it puts pressure on the outside of the heart such that blood coming in from the veins cannot enter the heart to be further circulated. The decrease in pump function happens incrementally, but certainly can get to the point where the heart no longer functions, blood no longer circulates, and the victim dies. A gunshot wound to the heart will also stop the pump from functioning, but along the same guidelines in terms of weapon power and caliber as previously discussed. A sniper’s bullet or firing squad would cause such massive disruption of the circulation that death would come quickly, but lower impact trauma may not even lead to death.
Now, what about bleeding to death? The larger the vessel, the quicker you bleed out, so anything that damages the aorta or the very large branches of the aorta will most quickly lead to death. However, the aorta is well protected. It comes off the heart behind the breastbone and then passes behind the heart and runs next to the front part of the spine. It is more vulnerable to injury when it passes below the diaphragm into the abdomen, but by then it has already branched off multiple times, so it is carrying a smaller (though still significant) amount of blood. If a killer wanted to kill someone by blood loss, they would need to go for the carotid arteries in the neck, the axillary arteries in the armpits, the abdominal aorta, or the femoral arteries. The carotids would be most effective, so long as the killer cut both of them — the brain would be most rapidly deprived of its blood supply (and ability to get rid of CO2), causing the victim to pass out, and without immediate care, the brainstem will die.
In the medical approach to trauma, we talk about the “golden hour.” When the injury doesn’t immediately cause death, there is a time (ranging from a few minutes to a couple of hours) where appropriate, intense medical intervention can truly save a life. In the ER, I have seen people survive stabbings to the heart, strangulations/suffocations, injuries to the aorta and carotids, punctured lungs, and all sorts of things that Hollywood portrays as instantaneous death.