A petaflop is the ability of a computer to do one quadrillion floating point operations per second (FLOPS). Additionally, a petaflop can be measured as one thousand teraflops.
A petaflop computer requires a massive number of computers working in parallel on the same problem. Applications might include real-time nuclear magnetic resonance imaging during surgery or even astrophysical simulation.
Today's fastest parallel computing operations are capable of petaflop speeds. The world's fastest supercomputer today, Titan, is capable of 20 petaflops.