It indeed used to be the case in the US that you couldn't hold dual citizenship (except in certain cases if you had dual citizenship from birth or childhood, in which case some Supreme Court rulings -- Perkins v. Elg (1939), Mandoli v. Acheson (1952), and Kawakita v. U.S. (1952) -- permitted you to keep both). However, most of the laws forbidding dual citizenship were struck down by the US Supreme Court in two cases: a 1967 decision, Afroyim v. Rusk, as well as a second ruling in 1980, Vance v. Terrazas.

Rules against dual citizenship still apply to some extent -- at least in theory -- to people who wish to become US citizens via naturalization. The Supreme Court chose to leave in place the requirement that new citizens must renounce their old citizenship during US naturalization. However, in practice, the State Department is no longer doing anything in the vast majority of situations where a new citizen's "old country" refuses to recognize the US renunciation and continues to consider the person's original citizenship to be in effect.

The official US State Department policy on dual citizenship today is that the United States does not favor it as a matter of policy because of various problems they feel it may cause, but the existence of dual citizenship is recognized (i.e., accepted) as a fact of life. That is, if you ask them if you ought to become a dual citizen, they will recommend against doing it; but if you tell them you are a dual citizen, they'll almost always say it's OK.

The problem arises when the a US citizen accepts citizenship in a country that requires them to give up US citizenship. For example, Ireland has no such requirement and so those who fulfill requirements for Irish citizenship but are born citizens of the US can hold citizenship in both Ireland and the US.

US law, on the other hand, requires naturalized citizens to renounce their previous citizenship.

Just another benefit of the decision to be born a US citizen....