It is the second amendment of the Constitution that provides this right. The first ten amendments are often referred to as the Bill of Rights, and so it would not be incorrect to say this is the second amendment of the Bill of Rights. Let's look at it here, and discuss it a bit:
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.
It is remarkable how much controversy has been created by these 27 words. The amendment has been interpreted by the Supreme Court on numerous occasions, it is the subject of much lobbying by gun advocates and those who advocate against guns, and it is issue that politicians are forced to deal with quite often.
What most people tend to not notice, or perhaps actively ignore, is that while this amendment gives the people the right to bear arms, it states quite specifically the reason for this. When we are interpreting the Constitution or any law, for that matter, it does not seem reasonable for us to ignore the stated purpose for the amendment or the law.
So, the argument that advocates against guns make is that the right to bear arms is meant to allow people to be ready if they must be called up to defend their country. And in fact, we have a "well regulated Militia" in the United States. It is called the National Guard. This is the militia of modern times, a group of people who maintain their training and stand by if they are called upon for a national defense of some sort. This means that there is no purpose in everyone else having any arms.
Those who argue for no gun control tend to ignore the stated purpose of this amendment, skipping over it until they get to the part about the right to bear arms. And in fact, most Supreme Court interpretations have agreed with their position.
Clearly, there is far more to each side than this, but as far as interpreting an amendment is concerned, these are the arguments that are made.