The Supreme Court has never officially ruled on the interpretation of the first clause of the 14th Amendment, which states: "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."
The interpretation of the section "subject to the jurisdiction thereof" is what leads to the birthright citizenship debate for illegal aliens.
However, Supreme Court Justices have written about the 14th Amendment in several Supreme Court opinions, explaining their interpretation of the law and their votes.
Here is a collection of Supreme Court opinions on birthright citizenship during the first decades after the 13th-15th Amendments were adopted. These opinions suggest that the amendments apply primarily to freed slaves.
"(N)o one can fail to be impressed with the one pervading purpose found in (the 13th, 14th and 15th Amendments), lying at the foundation of each, and without which none of them would have been even suggested; we mean the freedom of the slave race, the security and firm establishment of that freedom, and the protection of the newly-made freeman and citizen from the oppressions of those who had formerly exercised unlimited dominion over him."
-- Supreme Court opinion in the Slaughterhouse cases (1873)
http://www.stetson.edu/law/faculty/bickel/civilrights/media/case-digest-slaughterhouse-cases.pdf
"[The 14th Amendment was] primarily designed to give freedom to persons of the African race, prevent their future enslavement, make them citizens, prevent discriminating State legislation against their rights as freemen, and secure to them the ballot."
-- Supreme Court opinion in Ex Parte Virginia (1879)
http://caselaw.findlaw.com/us-supreme-court/100/339.html
"The 14th Amendment was framed and adopted ... to assure to the colored race the enjoyment of all the civil rights that, under the law, are enjoyed by white persons, and to give to that race the protection of the general government in that enjoyment whenever it should be denied by the States."
-- Supreme Court opinion in Strauder v. West Virginia (1880)
https://www.law.cornell.edu/supremecourt/text/100/303
"The right secured to the colored man under the 14th Amendment and the civil rights laws is that he shall not be discriminated against solely on account of his race or color."
-- Supreme Court opinion in Neal v. Delaware (1880)
https://www.law.cornell.edu/supremecourt/text/103/370
"The main object of the opening sentence of the 14th Amendment was ... to put it beyond doubt that all persons, white or black, and whether formerly slaves or not, born or naturalized in the United States, and owing no allegiance to any alien power, should be citizens of the United States ... The evident meaning of (the words, "and subject to the jurisdiction thereof") is, not merely subject in some respect or degree to the jurisdiction of the United States, but completely subject to their political jurisdiction, and owing them direct and immediate allegiance. ... Persons not thus subject to the jurisdiction of the United States at the time of birth cannot become so afterward, except by being naturalized..."
— Supreme Court opinion in Elk v. Wilkins (1884)
https://www.law.cornell.edu/supremecourt/text/112/94
There have been later rulings that dealt with birthright citizenship for legal immigrants such as the case in United States v. Wong Kim Ark (1898). This case is often cited by supporters of Birthright Citizenship, but the Court clearly ruled that the defendant's parents were legal permanent residents of the United States at the time of his birth, and not once was the legal status of the parents or the defendant ever questioned.
The Supreme Court ruled:
"[A] child born in the United States, of parents of Chinese descent, who at the time of his birth, are subjects of the emperor of China, but have a permanent domicile and residence in the United States, and... are not employed in any diplomatic or official capacity under the Emperor of China, becomes at the time of his birth a citizen of the United States."
https://www.law.cornell.edu/supremecourt/text/169/649
