The USA is one of the 63 countries whose citizens enjoy visa-free entry into the EU Schengen zone along with the UK and associated states like Iceland, Liechtenstein, Norway, and Switzerland.
This means that if you are an American citizen with a valid US passport and want to travel to Europe for a limited amount of time for tourism or business purpose, you can do so anytime you want to. However, if you have more long-term plans (say, you intend to get employed, study, or simply live in Europe for an extended period of time), you will need to apply for a visa. The type of your visa and required documentation depend on your goals.
Update for 2022: EU is set to launch ETIAS (European Travel Information and Authorization System) by the end of this year. It is similar to ESTA and is supposed to help simplify travel procedures and decrease security risks. All travelers allowed to travel to the EU and associated states visa-free will require an ETIAS authorization. This also includes US citizens.