So the issue is debatable.
1. Is "natural born" a separate class of citizenship or just an extra qualification to be president?
The argument is that there are only born citizens and naturalized citizens, and that born citizenship is synonymous with natural born citizenship.
2. Is natural born citizen the same as natural born subject in English law?
This debate attempts to trace the common understanding of the term at the time. The assumption is that since it was not defined explicitly in the Constitution, it was a term that was so commonly in use as to not require definition. Sources are Blackstone, Vattel, et. al.
Often, this debate descends into the importance that the Framers had copies of Vattel in their library, and writings that they referenced them. Also, the Constitution contains the phrase "Law of Nations."
This is where I cite Thomas Paine, because it is the only contemporary writing of the time (that I could find) that says that the President must not be a foreigner or half a foreigner.
Others cite the debates around the 14th amendment, which took place close to 100 years later and is too far removed from the time of the Framers to show original intent.
3. Is one parent enough to confer citizenship? Does birth have to take place in the country?
This is the jus sanguinis and jus soli argument. This goes back to Blackstone and Vattel.
Rebuttals cite Chester Arthur, who may have had a Canadian father and tried to hide it, evidenced by burning all of his records after his presidency. Others call out Mitt Romney's father and whether he was a Mexican citizen or not. Then McCain's birth in the Panama Canal Zone (or just outside of it) is thrown in for good measure.
4. What about children who are born to soldiers who are overseas? What about orphans whose parents are unknown? What about anchor babies? What about tourists who give birth while they are here?
These are emotional arguments. I fall on the side that says that life is not always fair. That said, the anchor baby and accidental tourist do suggest that they were never intended to be natural born citizens, so just being born here is by itself not enough to be natural born, especially if neither parent is a citizen.
5. There is the racism debate. Some say that this is only an issue because Obama is black. Others say that the lineage of all the past presidents was never in doubt, so testing the natural born requirement was never needed before now.
6. There is debate over the meaning of past Supreme Court cases on the matter.
The two most often cited are USA v. Wong Kim Ark and Happersett v. Minor.
Debate often swirls around majority vs. minority opinion, and dicta.
Compounding this debate is the fact discovered in 2011 that someone tampered with the archives at Justia.com just before the 2008 election and removed all the Supreme Court references of these cases from other Supreme Court cases, making it very hard for someone researching this to form a full conclusion. These references were restored after the election was over. This suggests that this topic worried someone of extreme influence.
-PJ