The Second Amendment says (emphasis mine):
A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed.
How are any laws restricting the "right of the people to keep and bear arms" constitutional?
Because the Supreme Court, who interprets the meaning of the Constitution, said so:
Like most rights, the Second Amendment right is not unlimited. It is not a right to keep and carry any weapon whatsoever in any manner whatsoever and for whatever purpose.
"District of Columbia, et al., v. Dick Anthony Heller. 554 U.S. 570". United States Supreme Court. June 26, 2008
Constitutional rights generally have limits and exceptions. The First Amendment, for example, reads:
Congress shall make no law ... abridging the freedom of speech, or of the press ...
But laws against obscene publications are constitutional, as are laws against defamation, "fighting words", copyright infringement, false advertisement, fraud, and revealing classified information. In each of these cases the conduct involves written or spoken words, but may still be prohibited without violating the First Amendment. (Although not always, as the "Pentagon Papers" case shows, for one example.)
Similarly, laws regulating gun ownership and possession to some degree have been held constitutional. For the matter of that, the Second Amendment was not generally considered to provide an individual right to gun ownership until the Heller case quoted in another answer, and some legal scholars still think that the Second should be limited to cases involving "a well-regulated militia".
Ron Trunk gave the right answer. For those who are confused about the meaning of "well regulated", see the full analysis (PDF) of the Second Amendment by Jeffrey Rosen, a law professor at George Washington University and President and CEO of the National Constitution Center, and Jack Rakove, the William Robertson Coe Professor of History at Stanford University. His book "Original Meanings: Politics and Ideas in the Making of the Constitution" won a Pulitzer Prize in History. (Emphasis mine in the following quotation.)
What did it mean to be well regulated? One of the biggest challenges in interpreting a centuries-old document is that the meanings of words change or diverge. "Well-regulated in the 18th century tended to be something like well-organized, well-armed, well-disciplined," says Rakove. "It didn't mean 'regulation' in the sense that we use it now, in that it's not about the regulatory state. There's been nuance there. It means the militia was in an effective shape to fight." In other words, it didn't mean the state was controlling the militia in a certain way, but rather that the militia was prepared to do its duty.
As explained in Justice Thomas's recent decision, the meaning of the constitution has to be interpreted in light of its historical context. If the people who wrote "shall not be infringed" nevertheless at the time routinely applied particular types of regulation to arms, then we can infer that their intention was limited in that way. Exceptions to the plain meaning of constitutional language have to be rooted in a historical tradition in which the rights in question were at the time understood to be limited. In the particular case of the 2nd amendment, Thomas observes that "The statutes essentially prohibited bearing arms in a way that spread “fear” or “terror” among the people, including by carrying of “dangerous and unusual weapons.”" We can therefore interpret this to say that while everyone has a right to responsibly bear arms for legitimate purposes like sport, hunting, self-defence, defence of the innocent, law-enforcement, and as a last ditch defence against tyranny, they don't (and never did) have the right to do so in a way that causes or threatens unjustified harm to others - threatening violence, intimidating, or for the purposes of committing crimes. Having a right to bear arms does not imply a right to do so in any manner possible, and the historical record shows that it was originally interpreted as excluding the use of arms in reckless, dangerous, or criminal behaviour.
Despite sweeping language in the Bill of Rights, the Supreme Court has consistently ruled that nearly every right can be subjected to restrictions. Most of those restrictions must satisfy at least a few criteria though.
First of all, almost any law has to satisfy a requirement that it be written to further some legitimate state interest. The minimum is that it must serve some "rational state interest". The strictest is that it must serve some "compelling state interest", and in between those is an intermediate level, that's applied primarily to cases involving discrimination against protected classes of people. A few cases have also talked about "heightened scrutiny", and it's not 100% clear whether that's exactly the same as intermediate scrutiny or not (but at the very least, it's pretty similar).
Assuming it meets that requirement (e.g., the state shows that it's for public safety) the next point to consider is whether it's tailored narrowly to infringe on rights as little as possible while accomplishing its goal.
So for example, let's consider when cars first started to come into wide use, and some people got killed in car accidents at night. The state has a legitimate interest in preventing that danger, so some sort of motor vehicle safety law passes the first hurdle.
A law requiring headlights when driving at night probably passes the second hurdle. A law prohibiting all cars probably doesn't.
Gun control laws have a pretty easy time passing the first hurdle. The claim is nearly always going to be that it's to protect public safety, and there's little question that'll be accepted as a legitimate state interest.
The second tends to be a lot harder hurdle for gun control laws to honestly meet. Oh, don't get me wrong. Some are obvious and easy. Prohibiting arms in prisons is obvious. Prohibiting an accused gangster from carrying a machine gun into the court room where he's on trial...again, pretty easy to support.
But what people normally think of as gun control laws? A field rife with battling statistics, and little room for a court to conclude with certainty that a particular law is really infringing on rights only as much as truly needed to maintain public safety.
I suppose I should add a couple of caveats.
First of all, a court only rules on a real case, so (for example) a lot of the old "blue laws" can remain on the books almost indefinitely as long as nobody's ever charged with breaking them, even though in a lot of cases it's pretty clear that they don't serve any legitimate state interest.
Second, over time, ideas of what constitutes a "legitimate state interest" change, often dramatically. 200 years ago, quite a few courts wouldn't have even questioned the "fact" that returning slaves to their owners was obviously a legitimate state interest. I feel fairly safe guessing that most people reading this today disagree. As much as we might wish otherwise, even supreme court justices are human, and influenced by the prevailing thought of the time, no matter how bad that may look in retrospect (e.g, Dred Scott).
A well regulated militia being necessary to the security of a free state, the right of the people to keep and bear arms shall not be infringed.
Third word. Regulations are inherently infringements, it's poorly worded when viewed with a modern lens.
It depends on who it is that's doing the restricting.
The promise of the second amendment was a promise made from the Federal Government to the states, not the other way around. With regards to the US Constitution, a state can place any restrictions that it wants. (However, most state Constitutions have their own guarantees.)
Simply because citizens of USA didn't care when they should have. When Thomas Jefferson worked on the amendment he wrote a lot about the problem. His principle was that citizens should be armed so that they are able to overthrow a bad government! In this light any state driven restrictions are absurd. Framers expected the goverment to go wild in less than 30 years requiring another violent revolution. It went wild, but no revolution happened - and that was only 30 years of 250...
But hey! Legal tender has to be gold or silver! Departments of health or education are plain unconstitutional as these areas are not listed as federal government's playground and the Constitution clearly says that any power not granted by the Constitution is not legal. Secret services are unconstitutional and that's why USA started with them as late as in 20th century. With Texas first joining exception no state apart from original 13 is Constitutionally a member state as there is no power of federal government to seize, buy, claim land and make it part of USA. And Porto Rico was seized by power in 20th century, thats no ancient history. The only way of getting a new member state is it's own voluntary will. Jefferson tried to make an amendment for Louisiana or a general one, but he was not sucessfull and he left most of principles behind: he started a war without congress aproval (don't blame Bush or Obama, they only repeated what this huge guy did), he put the federal army on steroids and expanded it about 20x (no other president made such a proportionally huge increase) This way we may continue, there is no single line in the Constitution that citizens did not let infringe. But at least you have something to return to, like Turkey has Mustafa Kemal. Most countries around the world have nothing, we have slavery in our Constitutions already without any need to build it gradually by overstepping tiny steps every now and then.
One argument is in what the Founders intended with the concept of "arms". At that time, it might include canons, pistols, and muskets. So the law could restrict weapons above this.
