Sources:

".U.S. law does not mention dual nationality or require a person to choose one citizenship or another. Also, a person who is automatically granted another citizenship does not risk losing U.S. citizenship. However, a person who acquires a foreign citizenship by applying for it may lose U.S. citizenship. In order to lose U.S. citizenship, the law requires that the person must apply for the foreign citizenship voluntarily, by free choice, and with the intention to give up U.S. citizenship." [1]

"Dual nationality can also occur when a person is naturalized in a foreign state without intending to relinquish U.S. nationality and is thereafter found not to have lost U.S. citizenship: the individual consequently may possess dual nationality. While recognizing the existence of dual nationality and permitting Americans to have other nationalities, the U.S. Government also recognizes the problems which it may cause." [2]

[1]: http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html

[2]: http://travel.state.gov/law/citizenship/citizenship_778.html

The US allows dual citizenship with any allied or non-hostile nations. US citizenship is only lost if the person expressly disavows US citizenship when or after becoming a citizen of another nation. (US and international law no longer recognize the concept of "nationless" citizenry.)

The US does not recognize dual citizenship with hostile nations. The last-acquired citizenship is deemed the person's current citizenship.

The consequence of dual citizenship is that you pay taxes, if applicable, to both countries as citizens. (Many countries have different tax rates for non-citizens.)

Edit: Follow-up to the above post -- The US treats its passports as superior to those of other nations, which is why you are urged to use your US passport when traveling in or out the US. You do not actually have to turn over your other passport. Israel, for example, requires citizens to use an Israeli passport for expedited re-entry.