An International Drivers Permit needs to be used in conjunction with a valid United States license. It's basically a translation of your existing drivers license into different languages, and provides some identifying information (photo, address, etc.). In the U.S. they can be obtain at AAA offices as well as from the National Automobile Club, typically for a fee of $15 or $20.
Some European countries require Americans to have an International Drivers Permit, while most do not. Many times, rental car companies will not requirement, but they may come in handy if you are pulled over for a traffic incident.
Countries that Require an IDP
It's best to check with the tourist board for the country you're visiting before you go, to get the latest information on exactly what you need to drive in another country. In general terms, though, most European countries do not require American drivers to have an IDP.
However, the following countries do: Austria, Bosnia-Herzegovina, Greece, Hungary, Poland, Italy, Slovenia, and Spain. Again, you might not even be asked for the IDP in these countries, but technically you're required to have one. Otherwise, you risk being fined.
The U.S. Department of State has good resources for overseas travelers, including country-specific road and traffic information. Their Road Safety Overseas page provides specific recommendations for safe driving.
What Out for Scams!
Travelers interested in International Drivers Permits should be aware of potential scams and outlets selling them for inflated prices. For more information, read my article on International Drivers Permits scams.