In the past citizenship was gained by the country you were born in. However, some of this has changed lately. For example, if a couple is a US citizen and now has moved to Canada, they had a child born in Canada, the child would have Candian citizenship and would also gain US citizenship because of their parents applying for this and the residency of the child when they move back to the US.
Another example is a man has a child from in another country and he is a US citizen he can apply for his child to become a US citizen even though the child was born in the Phillippines and lives in this country with his birth mother.
If an American citizen moves to another country such as France and has lived there for an extended number of years decides to apply for and is accepted for their French citizenship, it is possible and there shouldn't be any issues with this.
However, if a person is a citizen of another country and now wants to become a US citizen, in the oath to become a US citizen they are to give up their rights of citizenship in the other country. If a person fails to do this there could be a loss of their US citizenship.
The US doesn't really like a person to have dual citizenship, but on the other hand, they really don't do much about this at all. I think the major issue, in this case, is if you are not a US citizen and want to become one, you are forced to give up your other citizenship when you become a US citizen. There doesn't seem to be much of an issue if a person was born in the US, lived there a good deal of their life, decided to move to another country and take up citizenship, the US doesn't really do much in these cases.