The Basics
AR-15 stands for Armalite Rifle, after the company that first manufactured the weapon.
The AR-15 is not a military weapon, as the military models and variants, known in the US as M-16 and M-4, have either one of two additional fire modes; Fully automatic and 3-round burst, in addition to Semi-automatic mode. This known as "select fire" mode by the US military.
Although the term "Assault Rifle" is commonly used to describe the AR-15, it is not, in the strictest sense of the term, an assault rifle as it is not used by any military in the world as a primary combat weapon.
The AR-15 is also commonly referred to as a "high powered" rifle, but this appellation is largely a misrepresentation of its power. The .223 (5.56mm) round used in the AR-15 produces a muzzle energy of ~1250 ft/lb at ~3130 feet per second (FPS). The round in question (there are variants that are slightly more and less powerful) was designed specifically to cause battlefield injury (not death) when fired from an M-16. The military generally prefers causing injuries on a battle field as it requires more personnel to leave the battlefield (the injured and whomever carries him). Most rifle rounds in general use are far more powerful than the .223.
A partial list:
.223 Remington/5.56 NATO: 1243 ft/l @ 3130 FPS
.308 Winchester: 2630 ft/lb @, 2700 FPS
7.62x51 NATO: 3000 ft/lb @ 3165 FPS (used in the M-14, predecessor to the M-16, similar to .308 Winchester).
.338 Lapua Magnum: 4768 ft/lb @ 2900 FPS (a .308 variant used by military snipers).
.50 BMG: 13,196 ft/lb @ 2815 FPS (Ten times as powerful as the .223). (1)
The .223 Remington/5.56 NATO cartridge was developed for the AR-15, so any AR variant (chambered in the same ammunition) can be considered to be functionally identical to the AR from a ballistics standpoint, with slightly lower or higher numbers caused by varying barrel lengths (shorter barrel=less power and FPS).
From a tactical standpoint, the AR platform is sufficient for combat operations (with the addition of select fire) with a maximum effective range of 550 meters and a high hit ratio on man sized targets at 300 meters with iron sights. The M-14 is considered accurate to 875 meters with optics.
The AR-15 is NOT a hunting rifle. Many states ban the use of any caliber smaller than 6mm because of the likelihood of prey being injured and then suffering unnecessarily as a result. The argument can be made for its use as a "varmint gun" but the truth is that it is a very effective anti-personnel weapon and its primary design purpose is for tactical use.
Gun Laws
Federal law gives us the NFA (National Firearms Act) of 1934, amended in 1986 (and other times) to prohibit the sale of any fully automatic firearm (machine gun) to any private person unless it is a legal transfer, approved by the ATF, of a previously registered (before May 19, 1986) and legally owned machine gun.
In 1994, the Federal Assault Weapons ban was enacted with a ten year expiration and not renewed in 2004. During this period, AR-15s and variants were not legally sold to private citizens in the US.
There is not a "gun show loophole", or there is a "gun show loophole" depending on how you look at it. The ATF has a rather long list of requirements in order to get a Federal Firearms License (FFL), including a "brick and mortar" location. The ATF will NOT issue licenses to gun show vendors unless they also have a retail location. You have to have an FFL in order to have access to the NICS (National Instant Criminal Background System). Less than half of US states (18 and D.C.) require all gun show vendors to perform background checks on firearm sales. What this means is that in states that do not require background checks many vendors sell firearms as "private" citizens rather than dealers, even if they sell hundreds or thousands of guns. In states that do require background checks, regular brick and mortar retailers often occupy the booths rather than "gun show only" vendors, but those retailers can't normally afford to compete in non-background check states with vendors who have virtually no overhead. Any licensed FFL holder that is also a gun show vendor must perform background checks regardless of state law permissions.
According to some studies (mostly discredited because of small sample size) as many as 40% of all firearms sales are "private" and completed without background checks. Of those, about ~20% are legitimate private sales, estate transfers, gifts, etc. A few points go to illegitimate private sales (think guy selling out of the trunk of his car) and the remaining ~80% are gun show sales. This means that the simple step of providing FFLs to gun show vendors would mean that we could potentially increase the percentage of sales using a background check from 60% to 90%. Among incarcerated gun criminals, the largest percentage report purchasing firearms from a "friend or relative".
I cannot overstate the importance of the last paragraph. A change in POLICY (not law) at the ATF could result in as many as 11.5 MILLION additional background checks and the accordant benefit of discouraging criminals from purchasing guns now easily available at gun shows.
The NICS is not very efficient. There is apparently very little sharing of information amongst government agencies and no access to terror watch lists and "no fly" lists. This should change. The system MUST be improved in order to truly work as intended by law. The "due process" argument about persons under investigation or on various terror lists are moot since in all cases, the person in question, by virtue of the investigation or list status, raises a reasonable suspicion that ownership of a firearm might be unlawful for them, therefore the background check should be suspended until such time as the underlying issues are resolved. Also, if you are under indictment for murder and attempt to buy a gun, you can, even though there is a strong possibility that you will not be eligible for that purchase in the near future. In this case there is also a reasonable concern, and that background check should also be suspended until the courts resolve the criminal case. More information needs to be put into the NICS database and doing so would not require any change in federal law.
Mental health professionals should also have access (and be required to submit data) regarding certain types of mental illness that may indicate a predisposition to violent or amoral behavior. Although this would appear to be a violation of doctor/patient confidentiality, it is only a peripheral precaution that is not shared with any other entities, such as employers, banks, etc, and it is the responsibility of the physician to protect the public from potentially lethal threats. It is essentially the same thing as reporting a deadly contagious disease to the CDC.
Conclusions
In 2010 (the most recent year for which I could find accurate figures) out of 11,078 homicides committed with a firearm, only 358 were committed with rifles.(2) That statistic speaks for itself.
It is far too easy to buy a gun without a background check, but the administration could fix that with a phone call to the ATF. The resistance to changing the constitution exists in part because states can (and often do) make rules that exceed the scope of the constitution and that is as it should be. Each state can choose to expand its gun legislation as it sees fit, and many already have, albeit (and unfortunately) to no great improvement in gun violence rates.
The purpose of the second amendment, regardless of what any idiotic internet memes say, is to ensure a well armed militia to combat tyranny in any form. This includes protecting us against our own government, if and when that becomes necessary. And before you start howling about "militias", a militia is any civilian fighting force or organization with a pseudo-military structure; in other words ANYONE can be a militia or organize a militia, the only condition being that they identify as militia. The second amendment is not for hunting.
It is stupid to say that the second amendment was written 250 years ago and only applies to flintlocks. If today's police and military (and now millions of bureaucrats from FEMA, TSA, IRS, etc) also had only flintlocks you could make that case. Otherwise, it is just an empty argument.
Any comparisons of gun violence rates in the US to other countries are inherently flawed and never tell the whole story. Japan, for example, has strict gun control (effectively an outright ban) and a very low gun crime rate. But that is only part of the story. They also have the most homogeneous society in the world and essentially ban most immigration. The crime rate is not just low for gun crimes, but across the board; white collar crime and minor crimes are significantly lower than in almost all other countries. Clearly, their culture has more to do with the crime rate than the availability of guns. Honduras, on the other hand, has the highest murder rate in the world. But they are also an unintentional conduit for drug traffic and are one of the poorest nations in the Americas. Gun ownership in Honduras is just 6 guns per 100 in population (compared to 42 per 100 in the US), and guns are strictly regulated. As you can see, there is no correlation between gun ownership and gun crime.
In the US, 67% of all homicides are committed with a firearm. 1 in 3 murders are committed with everything from baseball bats to high heeled shoes. The lesson here is that if someone wants to kill you, they don't always need a gun to do it.
Mass shootings (commonly classified as 4 or more victims), although on the rise still represent only a tiny percentage of overall gun deaths. Mass shootings are very likely to increase as the hijra (jihad by migration) continues to increase here as it has in Europe.
Gun laws do not need to be re-written. The existing laws need to be enforced and policies need to be examined and improved to ensure universal compliance with background check requirements and conditions of gun ownership, including mental health status and pending legal actions.