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.