The 14th Amendment defines citizenship this way: "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside." But even this does not get specific enough. As usual, the Constitution provides the framework for the law, but it is the law that fills in the gaps.

Currently, Title 8 of the U.S. Code fills in those gaps. Section 1401 defines the following as people who are "citizens of the United States at birth:"

Anyone born inside the United States
Any Indian or Eskimo born in the United States, provided being a citizen of the U.S. does not impair the person's status as a citizen of the tribe
Any one born outside the United States, both of whose parents are citizens of the U.S., as long as one parent has lived in the U.S.
Any one born outside the United States, if one parent is a citizen and lived in the U.S. for at least one year and the other parent is a U.S. national.
Any one born in a U.S. possession, if one parent is a citizen and lived in the U.S. for at least one year
Any one found in the U.S. under the age of five, whose parentage cannot be determined, as long as proof of non-citizenship is not provided by age 21
Any one born outside the United States, if one parent is an alien and as long as the other parent is a citizen of the U.S. who lived in the U.S. for at least five years (with military and diplomatic service included in this time)
A final, historical condition: a person born before 5/24/1934 of an alien father and a U.S. citizen mother who has lived in the U.S.
Anyone falling into these categories is considered natural-born, and is eligible to run for President or Vice President. These provisions allow the children of military families to be considered natural-born, for example.