Is it possible to be a dual citizen of the United States of America and another country?
YES -- in many cases.
If you have been a dual citizen from birth or childhood, or else became a citizen of another country after already having US citizenship, and the other country in question does not have any laws or regulations requiring you to formally renounce your US citizenship before US consular officials, then current US law unambiguously assures your right to keep both citizenships for life.
The US State Department -- once quite combative in its handling of dual-citizenship claims -- has changed the way it handles these cases in recent years, and it is now much easier to retain such a status without a fight than it used to be.
The situation is slightly less clear for someone who becomes a US citizen via naturalization and still wishes to take advantage of his old citizenship. People who go through US naturalization are required to state under oath that they are renouncing their old citizenship, and conduct inconsistent with this pledge could theoretically lead to loss of one's US status.
However, the State Department is no longer actively pursuing cases of this nature in most situations. In particular, when a new American's "old country" refuses to recognize the US naturalization oath (with its renunciatory clause) as having any effect on its own citizenship laws -- and insists that the person in question must continue to deal with his old country as a citizen thereof (e.g., by using that country's passport when travelling there to visit) -- the US State Department generally no longer minds.
Similarly, the State Department doesn't seem to be doing anything any more to people who renounce their US citizenship as part of a foreign country's "routine" naturalization procedure (in a manner similar to what the US makes its new citizens do). However, if the other country in question requires its newly naturalized citizens to approach officials of their old countries to revoke their previous status, one will generally not be able to remain a citizen both of that country and the US.