According to Section 101(a)(22) of the Immigration and Nationality Act (INA) US National means
A Citizen of the United States
A person who may not be born in the US but holds permanent loyalty to the US.
There is a term called Dual Nationality which means a person can have the nationality of 2 countries at the same time and not having to forego nationality of one nation in pursuit of citizenship of the other country. For example, a person born in the US may have a citizenship of a foreign country without any danger to the current US citizenship. A person can have these advantages sometimes by law structure of a particular country and not really because he wished to.
There is also one term called Non-Citizen Nationality. These are basically the children born in America Samoa or Swains Island to the biological parents who were never the US citizens.
The US never mentions dual citizenship term or asks to give up the current nationality in order to be a US national. For example, a person born in the US may have a citizenship of a foreign country without any danger to the current US citizenship. But if a person is above 18 and has applied for foreign nationality, and he wishes to give away his US nationality, he can do so.
The person who wishes to give away US nationality for foreign nationality must do so without any pressure and it should be a voluntary decision. His records, statements, character, conduct should indicate that he wants to give up US citizenship by his own choice and not under any pressure by anyone.
Dual nationality though may seem good but sometimes also create some circumstances where a person is held up. As a dual citizen, the person has the obligation to abide by the rules of both the nations. In some cases, laws of one country may conflict with the rules of US and thus the US may end up not providing any legal protection to you.
Just like any other country, US wants its passport to be used to enter and leave the country for a dual citizenship national. In the same way, other country demands the same.