Citizenship is most commonly acquired by birth, marriage, or naturalization.
When immigrating to a different country and taking up citizenship through e.g.
naturalization, you become an individual with dual citizenship. Multiple or dual
citizenship is not something one can "apply for". Every country makes its own
rules as to who its citizens will be.
When being a dual citizen, you are bound by the rules of the local country
you actual reside in. For example, an Indian citizen who moves to the United
Kingdom and takes up U.K. citizenship must adhere to local rules and regulations
pending on what country he or she stays in. The Indian individual does now have
Indian rules to abide by when visiting India and U.K. rules to follow when
residing in the U.K. Citizenship is a complex issue, and it is important that
foreign citizens that wish to immigrate to the European Union is aware of their
own and new rules and regulations for dual citizenship.