The "peak inverse voltage", or reverse voltage, is what is will stand up to before dying. It will normally die quietly in this mode, just turning itself into a plastic blob totally indistinguishable from what it was before.
The "forward voltage" is the voltage that you can measure across it when it is in circuit and conducting. The available voltage will normally be greater than this, but there will be something in there to limit current. Sometimes this will be a resistor or other current limiting circuit, sometimes reliance will be placed on the internal resistance to limit current. If an LED tries to pass too much forward current, it will respond by dying, in this case usually with a very bright flash.