PAP, or postactivation potentiation, is basically the enhancement of performance in explosive, powerful movements following a heavy resistance training exercise (for the same muscles involved).
The idea is this: As an example, we will use a 100 meter sprinter.  After a heavy lift involving muscles that will be used in the sprint (barbell squat) the central nervous system is highly stimulated, resulting in greater and more motor unit recruitment in those muscles following that lift (i.e. more muscle fibers working for you in the sprint).  What follows is a much faster contraction rate and tension development in those muscles.
Does it work?  Almost all current research suggests that it does.  The most recent 5 studies on PAP showed significant improvements in times for swimmers, cyclists, and sprinters.  The study involving sprinters in particular showed an improvement in performances ranging from 0.89-3.3%, which is usually the difference between first and last on a fast track.
There is no doubt that PAP is a viable warm-up tool for power athletes.  The questions now are: which exercises and how much rest before the event? 
PAP benefit has been shown to last between 3-20 minutes in tested athletes, with the average recovery time between the lift and the event being 8 minutes.  Of note here is that some athletes took longer to recover, showing fatigue when not rested enough, and some recovered much quicker.  Each athlete should determine their own recovery time prior to a competition. 
Exercise selection, as noted earlier, should involve muscles that are involved in the actual sports movement itself, and should be compound (involving the coordination of several joints). 
Further research is needed to entirely understand the benefits of PAP, but the potential for improved performance is definitely worth considering PAP in a pre competition warmup.
Matt Pirtle, CSCS
Emerge Fitness Training
St. Charles Personal Training