The congress shall have the power...To establish an uniform Rule of Naturalization.
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
8 U.S. Code § 1154
Which has a whole load of laws passed by congress defining who can be called an immigrant.
8 U.S. Code Chapter 12, Subchapter II - IMMIGRATION
Which lays out all of the laws regarding immigration.
8 U.S. Code § 1181
Which blocks entry except in certain cases.