An arrow, in its simplest form, is a pointed stick fired from a bow. This is neither particularly accurate or dangerous, so over the years, many improvements have been made to the basic design. For no particular reason other than because it makes sense to me, I will start at the back and work forward.

This is probably the first, and most important improvements in arrow design, and has stood the test of time. Little more than a groove on the butt of the arrow, the notch serves to hold the arrow on the string. I imagine this was probably figured out fairly soon after the idea of the bow and arrow, as it is nearly impossible to fire an unnotched arrow. Notching on modern arrows differs little from notching on older arrows, except where materials technology has allowed the notches to be made better (more durable, smoother, etc)

Fletching is the term for the fins at the end of the arrow, meant to stabilize the flight. The earliest fletches were slats of wood, but this was soon switched to feathers, as wood created too much wind resistance, and weighed down the end of the arrow. Arrows typically had three or four fletches, 120o or 90o apart respectively.

The fletching in modern arrows differs greatly from older fletching in many ways. Feathers are rarely used, but this is more to avoid killing birds, and to avoid the inconsistencies of organic material. Plastics are now the most common material, either in the form of fins, or imitation feathers.

Another modern development is the idea of spiral fletching. Much like rifling the barrel to spin the bullet increases accuracy, the spiral fletchings spin the arrow, stabilizing the flight even further. This lead to some changes in arrowhead design, to be discussed below.

Shaft (yer damn right)
The shaft is fairly simple, and has changed little except in the materials area over the years. A good shaft should be straight, light, and strong. It should also be somewhat flexible, otherwise it will shatter on impact, and that’s just wasteful. Wood was the favored material for centuries, eventually replaced by fiberglass. The length of the shaft, along with the draw of the bow, was a determining factor in power. A longer shaft could be drawn back further without falling off the bow than a shorter one.

Pardon the pun, but now we get to the point. In the beginning, the arrowhead was just the end of the shaft, sharpened. While this would be useful for hunting vampires, it has a couple major drawbacks. The first is that wood can only be sharpened so much, and that its too soft to hold a good point for any length of time. The second is that an arrow that is fletched, with nothing but a pointed shaft for a head, is weighted towards the butt, and much less likely to hit with the point.

Early arrowheads were pointy bits of stone, tied to the end of the shaft. In areas where metalworking wasn't developed (like the aboriginal people of North America) this was eventually perfected, with chipped flint arrowheads being made that were sharper than today’s surgical scalpels.

On the Eurasian continent metalworking led to many improvements in arrowhead design. The most common were the broadhead arrowheads, which were triangle shaped blades, with a narrow tang at the base to attach it to the shaft.
Common variations were barbs to make them more difficult to remove, narrower points to increase penetration, or wider points to make larger wounds.

Some exotic arrowhead designs emerged, including blunt, cone shaped heads meant to stun, or at least bludgeon, and "frog-cutters", blades whose striking area was the inside of a "V", meant mainly to cut ropes (if this seems unbelievably difficult, keep in mind that the frog-cutters are a Japanese invention, and they produced some of the most frightening archers in history. See also Zen Archery)

Back to the standard broadhead. These generic arrowheads came in two main types - vertical and horizontal, each with its own purpose. Vertical broadheads had the arrowhead running parallel with the notch, meaning that the arrow would be fired with the arrowhead vertical. While the fletching could not be counted on to keep the arrow from spinning all the time, odds are an arrow would hit with the arrowhead in the same alignment it was in when it was fired. The ribs of most, if not all, quadrupedal animals, particularly those hunted for food, run vertical as well. This means for an arrow to have the best chance of getting past the ribs to the vitals, its head must too be running vertical. Vertical broadheads were designed for hunting with this in mind.

Now, if one type of arrow is for hunting, what was the other main use for arrows? If that isn't obvious enough, consider this: vertical arrows penetrate vertical ribs, so horizontal arrows penetrate horizontal ribs, which are most commonly found in humans. That’s right folks, horizontal broadheads were meant for war, designed specifically to kill other humans.

This is why you sometimes see characters in movies holding the bow horizontally to fire. Its not some kind of medieval gangsta-chic, but a case of trying to do the right thing with the wrong arrow. The character has most likely found himself stuck with nothing but hunting arrows when human-killing is the order of the day. Firing the bow in this position lessens the power of the shot, as you can't draw as far with your body in the way, but it beats having your arrow stopped by a rib.

As I said before, the invention of spiral fletching brought about a change in arrowhead design. With the arrow spinning, the orientation of the head was no longer an issue. Modern arrowheads are typically razor pointed shafts, with three or four triangular razor blades jutting out from there. Giving up on trying to slip between the ribs, these bad boys just go for the nastiest wounds they can. Barbs are very common.