What happened in the US vs. Miller case?
In this case, the Supreme Court ruled that a sawed-off shotgun was not protected by the Second Amendment.
The Second Amendment reads
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.
In the Miller case, the Supreme Court ruled that this meant that Miller and his friend only had a right to possess such weapons as could reasonably be used for military defense. In other words, the Court ruled that the whole point of the Second Amendment was to allow for the maintenance of a "well regulated militia." Outside of that context, it ruled, there is no right to own or carry weapons.
The Court ruled that the sawed-off shotgun had no reasonable relation to the maintenance of a militia. Therefore, the government could ban possession of such a weapon.