The Second Amendment reads, in full, "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."
Security in this sense means "safekeeping, defense, and protection." Infringed simply means "to trespass or violate"; in this case, the right of the people to keep and bear arms shall not be violated.
As simple as these definitions seem, how they are used and how they can be interpreted have been the subject of great and ongoing debate. Because English grammar and usage have changed in the two-plus centuries since this amendment was written, exactly what this amendment means has been interpreted differently by different sides.
First of all, the placement of commas in the Second Amendment makes for strange reading by today's standards. A modern translation might be, "Because maintaining a well-regulated Militia is necessary for the security of a free State, the right of the people to keep and bear Arms shall not be infringed." Now, what exactly is it that ought not be infringed?
Those who oppose gun control laws point out that, at the time the Second Amendment was written, militias were not groups of professional soldiers — like the Armed Forces are today — but were made up of the common citizenry. When trouble arose and the militias were called up, able-bodied men were expected show up with their own privately purchased weapons and do their duty. The Second Amendment was therefore intended to protect each individual's right to own firearms for the protection of clan and country.
People who favor gun control laws disagree. Gone are those days of local militias, having been replaced by the armed forces, including the army reserves and the coast guard. In these modern times, they say, militia refers to these armed forces. Futhermore, the phrase the people in this amendment doesn't refer to individual citizens, but to the U.S. population as a unit, much in the way that "We the People of the United States of America . . . " refers not to a bunch of individuals, but to the American people as a single whole. Therefore, the Second Amendment shouldn't be interpreted as protection of individuals' right to own firearms, but to the right of the federal government to maintain formal, organized militias like the Air Force and Marines. Therefore, they argue, laws that limit a common citizen's ability to own firearms do not violate the second amendment.