Dual citizenship (multiple citizenship) simply means that an individual is a citizen of two countries at the same given time. One can also be a citizen of three or more countries. However, all countries have their own laws regarding the same. Some countries allow it and some do not, while certain countries have no particular laws regarding this.
This process is not something that can be applied for but that happens when a person becomes a citizen of another country, in addition to his/her country of birth. Certain individuals get multiple citizenship automatically. For example, a child born in the US to foreign parents gets dual citizenship as it is automatically a citizen of the US and also a citizen of its parent’s home country. This also applies to children of US citizens born abroad where the child is both a US citizen and a citizen of the country of birth.
The Immigration and Nationality Act (INA) does not clearly define dual citizenship and it does not take a position for or against it. There has been no restrictions against such citizenship, but some provisions of the INA and earlier US laws were designed in such a way that that it reduces situations in which dual citizenship exists. Although it is mandatory for naturalizing citizens to undertake an oath renouncing previous allegiances to other countries, the oath has never been enforced for the actual termination of original citizenship.
Though the US Government does not endorse dual citizenship, it does recognize its existence and completely accepts the maintenance of multiple citizenship by US citizens. In the past, claims of other countries on dual-national US citizens sometimes put them in situations where their obligations to one country were in conflict with the laws of the other country. But, as fewer countries need military service and most base other obligations, such as the payment of taxes, particularly on residence and not citizenship, such conflicts have reduced. So recently, there is a significant increase in the number of persons who maintain their US citizenship in some other country.
If you take up citizenship in another country, then the US will consider you no longer be a citizen. However, the same is not necessarily true the other way around. If you are from another country and become a US citizen then, depending upon the laws of your country, that country may still consider you to be a citizen. 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, being a citizen of another country and if you are going ahead with the citizenship process with the US, you have to file Form N-400, Application for Naturalization with the USCIS.