"Pay it forward" means that one does a kind act for another person, a recipient, without asking for anything in return. Instead of requesting a form of payback, the first person instructs the recipient to "pay it forward" by doing the same for another in need.
This phrase gained recognition following the movie, "Pay it forward" starring Haley Joel Osment and Kevin Spacey. However, the concept had been around prior to the making of the movie. A reference can be found in the 1951 book "Between Planets" by Robert A. Heinlein. Nonetheless, the idea remains the same: that one continues the cycle of selflessness by being the first to commit an act of kindness or by following the lead of someone else who commits an act of kindness.