Well, a pi system is one in which you have a series of electrons that are in p orbitals. So basically, sp3 atoms cannot do this while sp2 and sp atoms can. The quintessential example is, of course, benzene. The p orbitals all undergo pi overlap and therefore they are "delocalized" in a ring made of p orbitals, i.e. the pi system.
With regard to peptide bonds, the lone pair on the nitrogen is in a p orbital because the nitrogen is sp2 hybridized (you should be able to tell orbital hybridization just by looking at it now). So, what it can do is donate into the pi antibonding orbital of the carbonyl (made from mixing the carbon and oxygen p orbitals), breaking the C=O pi bond and forming a C=N pi bond. This results in resonance delocalization, which stabilizes the amide. However, it should be noted (since I am a chemist) though perhaps beyond the scope of the MCAT, that breaking an amide bond is relatively simple, meaning that the resonance form is not a huge stabilizing force. Kicking out the amine is the RDS of peptide hydrolysis. So the resonance stabilizes the amide but not by that much.