Tomoya Kawakita was a dual US/Japanese citizen (born in the US to Japanese parents). He was in Japan when World War II broke out, and because of the war was unable to return to the US. During the war, he actively supported the Japanese cause and abused US prisoners of war who had been forced to work under him. After the war, he returned to the US on a US passport, and shortly thereafter he was charged with (and convicted of) treason for his wartime activities.
Kawakita claimed that he had lost his US citizenship by registering in Japan as a Japanese national during the war, and as a result he could not be found guilty of treason against the US. Presumably, the reason Kawakita fought so tenaciously not to be considered a US citizen was that he saw this as the only way to escape a death sentence for his treason conviction.
However, the Supreme Court ruled that since Kawakita had dual nationality by birth, when he registered himself as Japanese, he was simply reaffirming an already existing fact and was not actually acquiring Japanese citizenship or renouncing his US citizenship.
The court acknowledged that a dual citizen, when in one of his countries of citizenship, is subject to that country's laws and cannot appeal to his other country of citizenship for assistance. However, even when the demands of both the US and the other country are in irreconcilable conflict -- such as in wartime -- a dual US/other citizen must still honor his obligations to the US even when in the other country.
Although Kawakita lost his appeal, his death sentence was eventually commuted by President Eisenhower. He was released from prison, stripped of his US citizenship, and deported to Japan.
The reason the respondent in this case (the second party named in the case's title) was the United States -- rather than a government official (such as the Secretary of Labor or the Secretary of State) -- is that the case started as a criminal prosecution rather than as a lawsuit.