The AR-15 is often referered to as a "Modern Sporting Rifle", which really means that it uses some relatively modern ideas in its design, which in turn lead to its somewhat unique look (but not that different from a host of others that follow the same principles). So what are these modern design principles?

Long guns (of which rifles are a subset) have been around for hundreds of years. Until fairly recently they all followed a similar design typified by this 1600's musket:

The two items of interest here are that the butt is much lower than the axis of the barrel. This is to raise the barrel level with the shooter's eye when the butt is placed against his shoulder so that he can aim by looking along the barrel, perhaps even with the aid of some primitive sights as in this example. The second item is how the shooter holds the gun, with one hand gripping the stock near the trigger and the other hand (usually the left) steadying the gun by gripping further along, under the barrel, with some sort of insulation (wood) to prevent that hand coming onto contact with the hot barrel. A hot barrel was not so much of a problem with a musket because it was a long time between shots so things cooled down, and you couldn't pour fresh powder into a hot barrel since it would ignite.

The same basic design continued to be followed, not really changed much by other technical changes that occurred such as metallic cartridges that saved the trouble of having to load powder and shot each time, bolt action to make insertion and removal of the metallic cartridge much easier, rifled barrels to improve accuracy, magazine feeding of the cartridges so that some number of cartridges could be pre-loaded and cycling the bolt ejected a spent case and loaded a fresh cartridge. This was typified by the Lee-Enfield rifle that was used by the British and Empire/Commonwealth forces in two world wars:

The next major advance in rifle design was the use of self-loading (or semi-automatic) principles. The first widely produced rifle of this design was the US Army M1 which has an internal magazine loaded from the top by a spring clip holding the bullets in a group.

So rifles have come a long way in their technology, but notice that they still follow the age-old design of sighting along the barrel, and so having the butt below the barrel axis.

One of the basic ideas of a long gun is that it is more accurate over a longer distance than the alternatives (pistol, or SMG such as the Thompson, MP5 etc.). One of the most critical aspects of accuracy is, of course, accurately aligning the gun with the target and holding it steady on target as the trigger is squeezed. This is, if you have never tried it, harder than it sounds.

The problem with this old layout is that the recoil from firing is along the axis of the barrel (obviously...). However, the point of contact with the shooter's shoulder, where this energy is going to be absorbed is below that. Because of this misalignment of forces there is a resulting upwards force that tends to flip the barrel upwards.

This was not a problem with muskets or single shot rifles because they have to be dismounted from the shoulder to re-load, then re-mounted and carefully re-aligned for the next shot. Even with magazine-fed bolt action rifles like the Lee-Enfield, most people can't really keep things aligned when operating the bolt. With the M1 there is no need to do anything, the rifle reloads itself, but all that careful work in alignment/sighting is usually lost as the rifle jerks upwards.

The designer of the AR-15 tackled this problem by having the butt on the same axis as the barrel:

Now all the force of recoil is aligned and delivered directly back to the shoulder of the shooter. In addition, since the stock is now at the same level and aligned with the barrel, the designer placed a buffer and spring within the stock to capture a lot of the energy of the bolt cycling and reduce the amount of recoil felt by the shooter significantly.

This design improves the ability of the shooter to keep the rifle on target for subsequent shots, but introduces a couple of other problems. First, the barrel is no longer at eye level, so the shooter cannot sight along it. This was resolved by raising the level of the sights. This is something of a compromise, since the sights are no longer in the same plane as the barrel. At close range the sights will be higher than the point of impact (the rifle will shoot low), so the sights have to be adjusted to sight slightly downwards, so meeting the axis of the barrel at some point (usually around 100 yds) the sights are only really accurate at this distance. This is complicated by the fact that bullets don't travel in a straight line anyway, but follow a curved trajectory, but that is a long story in itself and left for another day.

The second problem is that the stock has moved upwards enough that it is no longer feasible to grip the stock and reach the trigger. This is addressed by the addition of a pistol grip which is used in place of the stock. 

Rather than an internal or fixed magazine the AR-15, and most modern rifles, uses an exchangeable magazine. This is sometimes seen as a questionable decision. The fixed magazine, top-loading design such as the Lee-Enfield had the advantage that it was possible to "top-off" the magazine after firing a few rounds. Its not possible to do that with the AR-15, its an all or nothing thing. The illustration above shows a 20 round magazine in place. This was actually the default configuration for the US M16 rifle (a full-auto variant of the AR-15) when it was first introduced in Vietnam. The larger 30 round magazine which replaced the 20 round version as the standard has the problem that it is larger. It's difficult to shoot prone with a 30 round magazine because it hits the ground, it makes the rifle deeper and slightly more difficult to maneuver.

The flash-hider on the tip of the barrel is not, as some people would like to have you believe, to hide the flash so that people can't see where you are shooting from, but to direct the burning gas forwards. Without this, it spreads out in a balloon of bright flame that destroys night vision if you are shooing in poor light or at night. The more modern versions of the AR-15 use some of the features of the M4 rifle:

The fixed stock is replaced with an adjustable stock, in recognition that people are different sizes, and that a correctly fitted stock is essential to accurate shooting. This has been known for many years, with very expensive high-end firearms (especially shotgun) manufacturers producing one-off, customized guns for people willing to spend a small fortune for a gun tailored exactly to fit them. The carry-handle which incorporated the rear sight has been replaced by a picatinny rail (an add-on carry handle, with rear sight can be attached to this for those that like that configuration) or you can add your own choice of rear sight, or even omit it and add an optical sight of some type (e.g. red dot). The front hand guard has been replaced with a tubular free-float hand guard which doesn't touch the barrel at all, it is fixed only to the body of the gun. Free-floating of the barrel can contribute significantly to accuracy. The hand guard usually incorporates some form of mounting system for attachments such as a flashlight, laser sight etc.