From the U.S. State Department.

The concept of dual citizenship (or dual nationality) means that a person is a citizen of two countries at the same time.  Each country has its own citizenship 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. citizen parents may be both a U.S. citizen and a citizen of the country of birth.

Generally, the U.S. government has looked with disfavor on U.S. citizens who maintain dual nationality.  In this regard, Congress has an appropriate concern with problems attendant to dual nationality such as allegiance and war-time support.  One who has dual nationality is potentially subject to claims from both nations, claims which at times may be competing and/or conflicting.  However, dual nationals owe allegiance to both the United States and the foreign country.  They are required to obey the laws of both countries.  Either country has the right to enforce its laws, particularly if the person later travels there.