Rather than pull quotes, if you've gotten this far in the thread, I'm just going to briefly echo some of the better sediments here. Additionally, I'll touch on some of the legal aspects of carrying and using a firearm which, for a trained gun owner, are at the forefront of thought every moment of every day. For our friends outside the U.S., and gun owners in the U.S. for that matter, I hope this serves as a brief idea of what responsible gun owners are like. We're not all rednecks over here, thankfully.
Firstly, it's been said quite a few times that police are trained to aim for center mass to minimize the risk of missing. This is true. The addendum to this is that any shooter, police or private, is legally (and morally) responsible for everywhere your bullet goes. If you're in your house, and you're forced to shoot an armed home invader, and you miss, your bullet is going to go through your wall, across the street, through the neighbor's wall, and hit little Timmy enjoying his cartoons. It does not matter if you had a good reason to shoot, if you kill little Timmy by accident, you are still a murderer, and legally liable.
If you have to draw, you must be aware of what's behind your target, and what's behind that. Now, there are ways to mitigate this issue. All bullets are not made alike, and obviously some kinds, including different calibers, are better for certain applications, but I'll touch on that in a second. For this reason, trained shooters, police and military included, are taught to aim for center mass to minimize the risk of what we call 'over-penetration'. Simply put, the chest and torso is the thickest part of a person, therefore presents the lowest risk of the bullet passing through the person and hitting something behind. A shot to say, the arm or knee, if you got really lucky and actually hit, the bullet would easy pass through the body and keep going, even if it hits bone, in many cases. Remember, that bullet is your responsibility from the time it leaves the barrel to the time it stops moving.
Second, type of bullet. Most everyone is aware of the two most common types, plain lead, and full metal jacket (FMJ. There also exists 'total metal jacket, which is slightly different, but for our purposes here, mostly the same thing) which is a lead bullet coated in a copper shell. Now, lead is a very soft metal. When used as a projectile, it starts spreading into a pancake and flattening the instant it hits something. The advantage to this is the round is less likely to keep traveling if you score a center mass shot, if it even leaves the body at all. FMJ rounds, by contrast, will punch right through a person, two walls, a mailbox, a tree, and hit another person. This is because the hard copper shell helps prevent the lead core from expanding and fragmenting.
In addition to those two, there exist such types of bullet, but by no means limited to, things like steel core, frangible, and hollow point ammunition. I use these as example because they are vastly different ends of the spectrum. Steel core is a FMJ round, but like the name implies, possesses a whole or partial steel core instead of lead. In short, more punching power. Usually found in rifle ammunition, it's most often used by the military, since they have a much higher expectation of encountering hostiles wearing body armor, but is civilian legal to own. Frangible rounds are interesting in that they are designed to fragment into such small pieces upon entering the first object they hit. They literally turn to sand after hitting a target. There are some interesting videos of people shooting a steel target at point blank range with frangible rounds, which would normally be a shrapnel hazard, but they're only getting peppered with sand. Good self defense ammo, but it's still rather expensive, and carries the additional problem of possibly not doing much damage to an assailant if he's wearing, say, heavy leather clothing or a kevlar vest.
Hollow point ammunition is a plain lead round, or lead ball, as it's called, with the front of the cap hollowed out. Hollow points are, in effect, designed to flatten out even faster than regular lead ball, while having the penetration power of a lead round. They are a good mix of punching power and lower risk of over-penetration, and as such enjoy the top spot on choice for self/home defense ammunition.
The reason why I explain all of this is that legally, your bullet choice can have repercussions beyond ballistic considerations. If you shoot an assailant, even if it was a legally justified shooting, or a 'clean' shoot, if you're loaded with steel core, you're going to get torn apart in court. The prosecution can and will paint you as whackjob out looking for an excuse to shoot someone, armed with ostensibly "military" ammo. Hornady, one of the better ammo manufacturers, makes a joke "zombie ammo". It is identical in all respects but two to their Critical Defense ammo, one designed for self defense carry. The differences are the color of the brass cartridge, and the color of the polymer the hollow point is filled with. That's it. If you use that ammo in a self defense shoot, though, the prosecution can (and has in cases already) make the jury think your a sociopath who dehumanized the person you shot by pretending he's a zombie. No shit, this has happened.
The point of all this is that there is a very, very long list of things that must be considered even before one starts carrying a firearm, and even more that have to be considered at the time it might be necessary to fire, in the heat of the moment. Specifically to the original topic question, firing center mass, and not to wound or disable, provides the best chance, in all cases, of the shoot going as intended, without complications and or unintended casualties.