Dual citizenship is basically when an individual is a citizen of two countries simultaneously. Some California residents might hold dual citizenship if they were born in the U.S. to immigrant parents or if they were born in another country to a citizen of that country and a U.S. citizen. Someone who has become a naturalized U.S. citizen can also seek to regain citizenship in his or her country of origin, which will cause the individual to have dual citizenship.
Someone with dual citizenship does have several advantages over someone who is a citizen of a single country, such as the ability to become a resident of either country without excessive legal complications. However, there can be drawbacks as well; someone with dual citizenship might have to pay taxes to two different countries, regardless of where the individual lives. Each country might also have residency requirements.
Certain actions can cause someone to lose his or her U.S. citizenship, including committing a treasonous act, formally renouncing citizenship or becoming a member of a hostile country's military. Fulfilling responsibilities as a citizen of another country will not cause someone to lose his or her U.S. citizenship. Some of these responsibilities might include serving in a public office, voting or carrying a passport from that country.
Despite not officially recognizing dual citizenship, the U.S. does not work against anyone who holds citizens' rights in two countries. If someone wishes to become a U.S. citizen while retaining any rights in his or her country of origin, a citizenship attorney could provide more information on how to do so. Such an option might be beneficial to someone who frequently conducts business in both countries or has family members who chose not to immigrate.
Source: Findlaw, "Dual Citizenship", August 18, 2014