Informally, a continuation allows the programmer to freeze the current state of a program and save this state. Then the program can be restarted from this point with a value of the programmers choice. One important aspect is that a continuation can be called multiple times, which makes it different from setjmp() and longjmp() of C fame.
Continuations has a wide number of appliances. From implementing threading systems, to hiding state in dynamic webpages. The latter of these being a really neat thing to do which can save the programmer from many hours of tedious writing.