Holding Dual Citizenship (being a US dual-national) means to be a citizen of two countries simultaneously. United States law is agnostic towards citizenship status in other countries, as U.S. law does not not mention dual citizenship. The United States does not require naturalizing citizens to renounce their foreign citizenship. U.S. nationals, including dual nationals, must use a U.S. passport to enter and leave the United States. They may travel on any passport they have otherwise.
While it is true that immigrants to the U.S. are required to take an oath of allegiance, where they will:
…renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty, of whom or which I have heretofore been a subject or citizen.
While this seems to indicate a "one person, one country" principle it does not in practice. In 1967 the U.S. State Department attempted to refuse to issue a new U.S. passport to a U.S. citizen who voted in an election in Israel. The Supreme Court ruled this violated the persons' rights and violated the Constitution. This decision overturned a law which read: “a person, who is a national of the United States, whether by birth or naturalization, shall lose his nationality by voting in a political election in a foreign state.” Therefore, dual-citizenship is sanctioned by the U.S. Supreme Court.
Of course there are problems a U.S. dual-national could encounter while abroad. Foreign countries may assert obligations upon their citizens which conflicts with U.S. law. US dual-nationals are nonetheless required to always comply with U.S. law. While abroad, the U.S. Government may find it more difficult to provide the consular protection enjoyed by Americans traveling abroad, particularly when in the country of their second nationality.
However, some countries may punish their citizens as criminals for exercising dual citizenship rights and/or not recognize dual citizenship. For example, in China, dual citizenship is not recognized and citizens automatically lose their citizenship upon gaining citizenship overseas.