Although the U.S. Supreme Court has never specifically addressed the meaning of "natural born citizen," there are several Supreme Court decisions that help define citizenship:

Dred Scott v. Sandford, 60 U.S. 393 (1857): In regard to the "natural born citizen" clause, the dissent states that it is acquired by place of birth (jus soli), not through blood or lineage (jus sanguinis): "The first section of the second article of the Constitution uses the language, 'a natural-born citizen.' It thus assumes that citizenship may be acquired by birth. Undoubtedly, this language of the Constitution was used in reference to that principle of public law, well understood in this country at the time of the adoption of the Constitution, which referred citizenship to the place of birth." (The majority opinion in this case was mostly overturned by the 14th Amendment.)

United States v. Wong Kim Ark, 169 U.S. 649 (1898): A person born within the jurisdiction of the U.S. to non-citizens who "are not employed in any diplomatic or official capacity" is automatically a citizen.

Weedin v. Chin Bow, 274 U.S. 657 (1927): A child born outside the U.S. cannot claim U.S. citizenship by birth through a U.S. citizen parent who had never lived in the U.S. prior to the child's birth. (This is still true today, although the specific statutes upon which the Supreme Court's ruling was based have changed since 1927.)

Perez v. Brownell, 356 U.S. 44 (1958): Although the 14th Amendment sets forth the two principal modes of acquiring citizenship (birth in the U.S. and naturalization), nothing restricts the power of Congress to withdraw citizenship. (This case was overturned by Afroyim v. Rusk.)

Montana v. Kennedy, 366 U.S. 308 (1961): A person born in 1906, whose mother was a native-born citizen of the United States and whose father was a foreign citizen, who was born overseas and then moved to the United States, was not a citizen of the United States by birth. (Note that the relevant laws have changed considerably since 1906, so this decision does not necessarily apply to later cases.)

Afroyim v. Rusk, 387 U.S. 253 (1967): The 14th Amendment's provision that "All persons born or naturalized in the United States . . . are citizens of the United States" completely controls the status of citizenship and prevents the involuntary cancellation of citizenship.

Rogers v. Bellei, 401 U.S. 815 (1971): A person who is born abroad to an American mother shall lose his or her citizenship unless he or she resides in this country for at least five years between the ages of 14 and 28. (This is no longer the case; the statute under which Mr. Bellei lost his citizenship was repealed by Congress in 1978.)

Vance v. Terrazas, 444 U.S. 252 (1980): Congress has the power to define acts of expatriation (i.e., loss of citizenship). However, intent to relinquish U.S. citizenship must be established specifically by a preponderance of evidence; such an intent may not be inferred automatically as a result of a person's having performed an act which Congress has designated as an expatriating act. However, when "one of the statutory expatriating acts is proved, it is constitutional to presume it to have been a voluntary act until and unless proved otherwise by the actor."

Miller v. Albright, 523 U.S. 420 (1998): A child born overseas to an American father and a foreign mother (not married) is not a U.S. citizen unless paternity is established before an established age (in this case 21). This case challenged the law on the grounds that U.S. law requires no explicit acknowledgment of parenthood in the case of a foreign-born child to an American mother and a foreign father (not married).

Nguyen v. INS, 533 U.S. 53 (2001): As in the Miller v. Albright case, the Court holds that a child born overseas to an American father and a foreign mother (not married) is not a U.S. citizen unless paternity is established before an established age (in this case 18). The child was brought to the U.S. before his sixth birthday and raised by his father; however, after a criminal conviction, deportation was ordered but the child claimed U.S. citizenship. His citizenship was denied because paternity had not been established prior to his 18th birthday. The Court upheld the law, once again affirming that Congress has the power to define citizenship outside the citizenship dictated by the 14th Amendment (citizenship by birth).
The Supreme Court, through case law, has created a guideline for U.S. citizenship. The following outlines the rulings of the Court:

The 14th Amendment completely controls the status of U.S. citizenship and prevents the involuntary cancellation of citizenship.
All persons born in the United States are citizens of the United States.
This applies to children born to legal and illegal residents.

This does not apply to children of foreign citizens employed in any diplomatic or official capacity.

Congress has the power to define acts of expatriation (i.e., loss of citizenship).
A person must voluntarily relinquish U.S. citizenship.

It is constitutional to presume it to have been a voluntary act until and unless proved otherwise by the actor.

Congress may revoke citizenship involuntarily if it has been obtained unlawfully.

Congress has the power to define citizenship outside birth in the U.S.

Congress can set different citizenship requirements for children born to American mothers versus American fathers.

Congress can require that U.S. citizenship must be established by a certain age for it to be recognized.

[link to en.wikipedia.org]