The Supreme Court of the United States has had only limited influence upon the meaning of this amendment over the years. As a result, we still don’t know exactly what the modern approach to this amendment would be.
The main case that sets forth some guidelines is United States v. Miller, 307 U.S. 174 (1939). This case is of questionable value, because the respondent, Miller, didn’t appear before the Court, having received a trial court determination that the law used to prosecute them violated the Second Amendment. For some reason, they saw no need to stick around and argue the (for them) theoretical correctness of that decision. 
In Miller, the Court held that the purpose of the Amendment was to ensure the ability to provide a well-regulated militia. Therefor, the second clause, the one about the actual right, had to be interpreted with this in mind. In the opinion of the Court, if the weapon in question had no reasonable relationship to the effort to preserve an efficient militia, the right to keep and/or bear it was not preserved by the Amendment. Amazingly enough, the Court didn’t think a sawed off shotgun fell into the category of protected arms.
The Supreme Court has not revisited this decision.
Now, there are some troubles with defining ‘arms’ this way. For one thing, do we regard the modern-day ‘militia’ as the equivalent of the National Guard (or something similar)? If so, would I be able to buy and old MIG or F-4 Phantom and claim my right to do so is protected by the amendment because it is reasonably related to preserving an efficient militia, which these days must include an air force of some sort, thus it is an ‘arm’ (by the way, as used, the term ‘arms’ is almost certainly a short form of ‘firearms’)? If a sawed-off shotgun wasn’t a protected weapon, what sorts of weapons today wouldn’t be protected under the same rationale (perhaps the average shotgun or deer-hunting rifle)? You can see how this seemingly simple definition from Miller quickly gets into trouble.
On top of this, there is underneath the definitional question the whole bubbling pot of discussion regarding the issue of whether the Amendment’s protection is an individual right to own firearms or a right reserved to the states to provide for a militia. Obviously, the Court in 1939 felt it was a state right at its most basic level. But this is hard to accept in light of the placement of the amendment squarely in the forefront of a whole list of amendments protecting individual rights. And while I am not going to open up the can of worms this issue presents (lest we be banished to the land of endless debating to no purpose), one can see that the definition of ‘arms’ would have to vary depending on the underpinnings of the right. A personal right to keep and bear arms would probably have to include the right to own a shotgun or rifle; both have some value to personal defense even today.
So, to answer the OP, the answer is, “No one knows for sure.” 