"I hereby announce, on promise, that I absolutely and entirely renounce and abjure all allegiance and reliability to any foreign prince, potentate, state or sovereignty of whom or which I have heretofore been a subject or citizen".
Some countries, like Canada, choose to ignore this renunciation and consider the new U.S. citizen is a citizen of Canada as well.
Why dual citizenship? For most people, it is irrelevant. For others it may be important because of different regulations on the right to work, vote, or have property rights in these countries seconds. Being a dual citizen may have beneficial effects or may create headaches double. Than being a dual citizen is right for you can not be answered by you and your situation. If you think you are a citizen of two countries is something that might work for you, check with a lawyer who specializes in this field. It is surely not a "do-it-yourself" plan.
List of Dual Citizenship Countries:
The United States of America
In addition to the countries mentioned above, the following countries also allow dual citizenship: Austria, Australia, Bangladesh, Belgium, Belize, Brazil, Colombia, Cyprus, Dominica, El Salvador, Finland, France, Germany, Grenada, Hungary, Iceland, Iran, Iraq, Ireland, Israel, Jordan, Latvia, Lithuania, Macedonia, Malta, Mexico, Montenegro, New Zealand, Pakistan, Philippines, Russia, Serbia, Spain (only in certain cases), Sri Lanka, St. Kitts & Nevis, Switzerland, Syria, Vietnam and Western Samoa.