That's if a person is naturalised, I think that most countries let a person be born as dual nationals, though. For instance, Americans demand that upon becoming a naturalised American, new citizens have to renounce all other nationalities (witness Anthony Hopkins). However I can keep my American nationality even after becoming naturalised to something else, too (?would take a constitutional amendment to change that position).
As it currently stands the oath you make to become a naturalised citizen of the US requires to to forswear your former nationality
"I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty of whom or which I have heretofore been a subject or citizen; that I will support and defend the Constitution and laws of the United States of America against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I will bear arms on behalf of the United States when required by the law; that I will perform noncombatant service in the Armed Forces of the United States when required by the law; that I will perform work of national importance under civilian direction when required by the law; and that I take this obligation freely without any mental reservation or purpose of evasion; so help me God."
The US does not technically recognise dual nationality for adults. If for example a US citizen became a citizen of another country - they would automatically loose thier US nationality as soon as they swore the oath to become a naturalised citizen of said country. The laws in the US are such that once you have done that they have the right to take you passport should they wish to do so. Many cases of Americans wanting to seek dual nationality to connect with a part of thier family have fallen fould of that (especially those seeking mexican / american dual nationality have fallen foul of this) but that is becasue as a young nation the US are aware of their need to bind thier citizens with oath and pledges. Kids here pledge their alliegance every day in school.
However if you are a forgeign national of a country with much odler laws this is where the grey area exists. With much older nations like our own and many other European nations, our laws are so old with regard to this that in order for a British Citizen to give up thier nationality and that to be legally recognised by Britain - you would have to (after having sworn your allegiance to the US or other country) go to the UK - and swear in court before a judge that you were no longer a British Citizen. Otherwise the crown does not recognise our right to simply swear to someone else that we give up our nationality. The US chooses not to enforce this and turns a blind eye to this loophole - which means that most British Citizens who have taken up US nationality are technically still able to use both passports quite legally. The US could choose at any time to change thier standing on this and insist that people legally forswear in thier parent nation as well but currently they choose not to. Anyone who decided to forswear thier British Nationality in a British court of law does this of thier own volition and not becasue they have been required to by the US. So although you would be perjuring youeself to do so you can become a US citizen whilst knowing that the crown doesn't give a rats what you've sworn and who to and you can be dual.
My DD is dual Brit / American - I would never want US citizen ship for any reasons, but my excuse has always been I'd rather not perjure myself and I am certainly never legally giving up my Britishness.....