1. Literally, “running around” [from Pennsylvania German].
2. A period when adolescent Amish explore the outside world before making an informed decision to either leave the Amish community (resulting in shunning) or be baptised as a full, adult member of the church.
Although the Amish do not approve of adolescent rebellion and breaks from church rules, during the period of Rumspringa, youth often engage in rebellious behavior contrary to the community’s strict standards. They can go on dates. They may secretly experience modern technology and music, drive cars, wear non-Amish clothes and hair styles. Some experiment with drinking, smoking, drugs and sex. In other words, during Rumspringa they see what the world outside the sheltered and stern Amish community is like, what it has to offer, before making a more informed decision about staying in the church.
More generally, then, Rumspringa is a period of exploring the world outside of your normal one, involving a radical break from habit and the comfortable safety of the familiar. It means breaking from tradition or custom, experiencing other life styles or communities, and basically exposing yourself to the larger world.
Note that the actual practice of Rumspringa among the Amish and consequences for misbehavior differ between communities and between individuals. For some it involves much less deviation from norms (often because small communities do not offer the anonymity found in larger ones), and generally sinning is still hidden from the community. For those curious about Rumspringa in the Amish, check out the awesome documentary Devil’s Playground, which NPR did a report on.