Continuations are constructs that are inspired from Scheme. Basically, they contain all information about a specific program location and the local method variables. This enables the application to create an interruption in program execution and continue later at the exact same location as if nothing happened.
Java's virtual machine doesn't have native support for continuations. RIFE however modifies your classes on-the-fly and inserts the required logic in an invisible manner. All regular tools such as debuggers continue to work, but you are now able to use Java language constructs to describe the flow of your web application logic.
Continuations can be hard to grasp, we've made a quicktime presentation (1.9MB) that explains the concept clearly.