Ilya Somin has already covered much of the Trump v. Barbara territory in his posting here on the VC a few days ago [available here], and I agree with everything he writes, including his terse summary: "The 6-3 decision was right, and a contrary ruling would have had horrific results."
I'll assume the basic framework of the case is familiar to you all. If not: The Citizenship Clause of the 14th Amendment says that anyone "born . . . in the United States and subject to the jurisdiction thereof" is a US citizen. The case turns on the meaning of those five italicized words: "subject to the jurisdiction thereof". Trump's Executive Order (#14160 [available here]) says that individuals born here whose mothers are present in the US "unlawfully" are not US citizens. He asserts that this does not violate the Citizenship Clause because those individuals are not "subject to the jurisdiction of the United States" within the meaning of that phrase – at least, within the meaning of that phrase in 1868, when the Citizenship Clause was added to the Constitution.
Plaintiffs, needless to say, disagree, as does a majority of the Court.