Might as well throw in an official "source" as well.
https://travel.state.gov/content/travel/en/legal/travel-legal-considerations/Advice-about-Possible-Loss-of-US-Nationality-Dual-Nationality/Dual-Nationality.html
Dual Nationality
Section 101(a)(22) of the Immigration and Nationality Act (INA) states that “the term ‘national of the United States’ means (A) a citizen of the United States, or (B) a person who, though not a citizen of the United States, owes permanent allegiance to the United States.” Therefore, U.S. citizens are also U.S. nationals. Non-citizen nationality status refers only individuals who were born either in American Samoa or on Swains Island to parents who are not citizens of the United States. The concept of dual nationality means that a person is a national of two countries at the same time. Each country has its own nationality laws based on its own policy. Persons may have dual nationality by automatic operation of different laws rather than by choice. For example, a child born in a foreign country to U.S. national parents may be both a U.S. national and a national of the country of birth. Or, an individual having one nationality at birth may naturalize at a later date in another country and become a dual national.
U.S. law does not mention dual nationality or require a person to choose one nationality or another. A U.S. citizen may naturalize in a foreign state without any risk to his or her U.S. citizenship. However, persons who acquire a foreign nationality after age 18 by applying for it may relinquish their U.S. nationality if they wish to do so. In order to relinquish U.S. nationality by virtue of naturalization as a citizen of a foreign state, the law requires that the person must apply for the foreign nationality voluntarily and with the intention to relinquish U.S. nationality. Intent may be shown by the person’s statements and conduct.
Dual nationals owe allegiance to both the United States and the foreign country. They are required to obey the laws of both countries, and either country has the right to enforce its laws. It is important to note the problems attendant to dual nationality. Claims of other countries upon U.S. dual-nationals often place them in situations where their obligations to one country are in conflict with the laws of the other. In addition, their dual nationality may hamper efforts of the U.S. Government to provide consular protection to them when they are abroad, especially when they are in the country of their second nationality.
U.S. nationals, including dual nationals, must use a U.S. passport to enter and leave the United States. Dual nationals may also be required by the foreign country to use its passport to enter and leave that country. Use of the foreign passport to travel to or from a country other than the United States is not inconsistent with U.S. law.
