Dual citizenship simply means a person is a citizen of two countries at the same time. Every country has its own laws for citizenship based on its own policy. Persons may acquire dual nationality by operation of different laws in countries rather than by choice. For example, a child born in a foreign country to US citizen parents can become both a US citizen and a citizen of the country of birth.
As a US citizen, you can acquire foreign citizenship through marriage, or if you are naturalized as a US citizen, you may not lose the citizenship of the country of birth. The immigration law in US does not mention dual citizenship or require a person to choose one citizenship or another. Additionally, if you are automatically granted another citizenship, you do not risk losing your US citizenship. However, if you acquire a foreign citizenship by applying for it, you may lose your US citizenship. In order to lose/renounce your US citizenship, per the law, you have to apply for the foreign citizenship voluntarily with the intention to give up US citizenship.
The US government recognizes that the concept of dual citizenship does exist but does not encourage it as a policy due to the problems that may rise. Claims of other countries on dual citizenship may conflict with US law. It may also limit US government efforts to assist citizens abroad. Although a naturalizing citizen has to undertake an oath renouncing previous allegiances to other countries, the oath has never been enforced for the actual termination of original citizenship.
The country where you, as a dual national are located generally has a stronger claim to your allegiance than that of the other country. Having said so, dual nationals do owe allegiance to both the United States and the foreign country. You have to obey the laws of both countries. Both the countries have the right to enforce its laws, particularly if you are likely to later travel there. Most US citizens, including dual nationals, are required to use a US passport to enter and leave the United States. You may also be required by the foreign country to use its passport to enter and leave that country.
Off late, there is a significant increase in the number of persons who maintain their US citizenship in some other country. Most countries allow a person to renounce/ lose citizenship. The process on how to renounce a foreign citizenship can be obtained from the foreign country’s embassy and consulates in the US. US citizens can renounce their US citizenship in the proper form at US embassies and consulates overseas.
It is important to note that not all countries allow multiple citizenship. If you are from a country that allows multiple citizenship with the US and similarly the other way round, you can apply for US citizenship.
Being a citizen of another country and if you wish to go ahead with the dual citizenship process with the US, you have to file Form N-400, Application for Naturalization with the USCIS.