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If you're going to be involved in government in the United States, citizenship is a must.
To be a Senator or Representative, you must be a citizen of the United States.
To be President, not only must you be a citizen, but you must also be natural-born.
Aside from participation in government, citizenship is an honor bestowed upon people by the citizenry of the United States when a non-citizen passes the required tests and submits to an oath.

Natural-born citizen

Who is a natural-born citizen?
Who, in other words, is a citizen at birth, such that that person can be a President someday?

The 14th Amendment defines citizenship this way:

and subject to the jurisdiction thereof,

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.
The Constitution authorizes the Congress to create clarifying legislation in
Section 5 of the 14th Amendment;
the Constitution, in Article 1, Section 8, Clause 4,
also allows the Congress to create law regarding naturalization,

Currently, Title 8 of the U.S. Code fills in the gaps left by the Constitution.
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:

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.

Separate sections handle territories that the United States has acquired over time, such as

Each of these sections confer citizenship on persons living in these territories as of a certain date,
and usually confer natural-born status on persons born in those territories after that date.
For example, for Puerto Rico, all persons born in Puerto Rico between April 11, 1899, and January 12, 1941, are automatically conferred citizenship as of the date the law was signed by the President (June 27, 1952).
Additionally, all persons born in Puerto Rico on or after January 13, 1941, are natural-born citizens of the United States.
Note that because of when the law was passed, for some, the natural-born status was retroactive.

The law contains one other section of historical note, concerning the Panama Canal Zone and the nation of Panama.
In 8 USC 1403, the law states that

Note that the terms "natural-born" or "citizen at birth" are missing from this section.

In 2008, when Arizona Senator John McCain ran for president on the Republican ticket, some theorized that

However, it should be noted that section 1403 was written to apply to a small group of people to whom section 1401 did not apply.
McCain is a natural-born citizen under 8 USC 1401(c):
"a person born outside of the United States and its outlying possessions of parents both of whom are citizens of the United States
and one of whom has had a residence in the United States or one of its outlying possessions, prior to the birth of such person."
Not everyone agrees that this section includes McCain - - but absent a court ruling either way, we must presume citizenship.

U.S. Nationals

A "national" is a person who is considered under the legal protection of a country, while not necessarily a citizen.
National status is generally conferred on persons who lived in places acquired by the U.S. before the date of acquisition.
A person can be a national-at-birth under a similar set of rules for a natural-born citizen.
U.S. nationals must go through the same processes as an immigrant to become a full citizen.
U.S. nationals who become citizens are not considered natural-born.

8 USC....is a law for Aliens.