As above, the best, and largest target is the heart/lung area. 99.9% of hits here (IMO) will result in a deer down inside 100 yards, often on the spot.
Head shots are best avoided. The reason is that it's possible to leave an animal with a broken jaw to starve to a horrible death. Deer can and often do, move their heads very quickly.
Also avoid any shot that will hit behind the diaphragm of the animal, the "gut shot" this inevitably results in a long tracking job, (can be many miles) to find, or not find an animal that will be a hell of a mess to clean out.
The so called "Texas Heart shot" (a shot taken from the rear with the objective of reaching the heart/lung area) while possible, is another shot to be avoided. it can result in an instant drop of the animal if you are using a rifle of sufficient power, with a bullet of construction to allow such penetration. However, like the above, it always results in a hell of a mess to clean up, and can result in a complete miss of the heart lung area, unless the animals body is completely in line with your shot, and the bullet stays true inside it's body.
The neck shot can have a lightning like kill effect too, however, like the head shot, it's an area of the deer that can move quickly while you are squeezing the trigger, resulting in a critical, or non-fatal wound, and a lost animal that may later die.
Running shots.
This is an area that can result in much argument among hunters, and here on the forum. Running shots, are completely possible, but you will have to figure the ethics out for yourself. If you decide to take one, I suggest you do not take any running shot beyond fifty yards. Preferably closer. Learning to lead a deer and not result in a gut shot is an art more than a science.
Better is to stop the deer, when you are ready to take your shot. This is done by a loud short shout, or a whistle, even by a deer call grunt if you can do so loudly. I say loud, because a running deer is often making enough noise that it may not hear you otherwise. I say short, because if you make a long noise, the deer will spot you. You want him to stop, to figure where the noise came from. Once I had to yell several times before the deer heard it. I yell woe! If he doesn't stop within about two of three bounds try again. BTW if the deer has seen you, probably all the yelling in the world won't stop it. Try anyway.
A walking deer can also be stopped in this manner, just don't yell.
Please respect the game you are hunting, and be responsible with your shots.