I didn't see the movie but I did read the book. It's by Catherine Ryan Hyde. The concept is what others have said -- let's say you do something nice for Person A. Instead of Person A paying you back, Person A pays it forward for doing something nice for Person B. And on and on.
Every so often you hear about someone at a toll booth who pays the toll for the car behind, who then pays it forward by paying the toll for the next car, who then pays the toll for the next car. That's a Pay It Forward chain.