I will defer to the US Supreme Court on this one:
The Constitution does not, in words, say who shall be natural-born citizens. Resort must be had elsewhere to ascertain that. At common-law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives, or natural-born citizens, as distinguished from aliens or foreigners. Some authorities go further and include as citizens children born within the jurisdiction without reference to the citizenship of their parents. As to this class there have been doubts, but never as to the first. For the purposes of this case it is not necessary to solve these doubts. It is sufficient for everything we have now to consider that all children born of citizen parents within the jurisdiction are themselves citizens. (Minor v. Happerset 88 US 162)
and
It thus clearly appears that by the law of England for the last three centuries, beginning before the settlement of this country, and continuing to the present day, aliens, while residing in the dominions possessed by the crown of England, were within the allegiance, the obedience, the faith or loyalty, the protection, the power, and the jurisdiction of the English sovereign; and therefore
every child born in England of alien parents was a natural-born subject, unless the child of an ambassador or other diplomatic agent of a foreign state, or of an alien enemy in hostile occupation of the place where the child was born. III.
The same rule was in force in all the English colonies upon this continent down to the time of the Declaration of Independence, and in the United States afterwards,
and continued to prevail under the constitution as originally established. (United States v. Wong Kim Ark 169 US 649)
A few other cases are listed on that bane of a data site - Wikipedia...
http://en.wikipedia.org/wiki/Natural_born_citizen