Rep. John Bingham of Ohio, considered the father of the Fourteenth Amendment, confirms the understanding and construction the framers used in regards to birthright and jurisdiction while speaking on civil rights of citizens in the House on March 9, 1866:
" ... I find no fault with the introductory clause [S 61 Bill], which is simply declaratory of what is written in the Constitution, that every human being born within the jurisdiction of the United States of parents [plural, meaning two] not owing allegiance to any foreign sovereignty is, in the language of your Constitution itself, a natural born citizen..." (http://americamustknow.com/default.aspx)
"For each presidential candidate, they can put the factual history of their birth in the equation and see if they fit the bill to be president of the U.S. in 2008 under Article II, Section 1, Clause 5 and the relevant federal law under U.S. v. Wong Kim Ark, 169 U.S. 649 (1898), and Perkins v. Elg, 307 U.S. 325 (1939). As you can clearly see, Obama is a U. S. citizen, but he's not a "natural born citizen" and, as such, is not eligible for POTUS, because his father, a Kenyan, was a foreigner. His birth certificate is only relevant to answer the question, "what does Obama have to hide?"
C H A R T at http://www.theobamafile.com/NaturalBornCitizenChart2.htm
U.S. v. Wong Kim Ark's importance is that it is the first case decided by the Supreme Court that attempts to explain the meaning of "natural born citizen" under Article II, Section 1, Clause 5 of the U.S. Constitution. Natural born citizen is similiar to the meaning of what a natural born subject is under Common Law in England. That is one of the reasons why the framers specifically included a grandfather clause (natural born Citizen OR a Citizen of the United States, at the time of adoption of this Constitution). The founding fathers knew that in order to be president, they had to grandfather themselves in because they were British subjects. If they didn't, they could not be President of the U.S. The holding in U.S. v. Wong Kim Ark states that Wong Kim Ark is a native born citizen. If you look at the fact of Wong Kim Ark being born in San Francisco, CA, that holding is correct.
Perkins v. Elg's importance is that it actually gives examples of what a Citizen of the U.S. is; what a native born American Citizen is; and what a natural born citizen of the U.S. is. A natural born citizen is a person who is born of two U.S. citizen parents AND born in the mainland of U.S.
You can go take a look at the excellent chart Beckwith (an attorney) has put together to explain differences in 'citizen'.
Fire For Effect. AKA FFE.