The flash point is the lowest temperature at which the fuel vapor can burn in the presence of a continuous ignition source. If the ignition source is removed, the fuel stops burning.
The fire point is the lowest temperature at which the fuel vapor will continue to burn after the ignition source is removed...much higher than the flash point.
Both of these are specification standards; they are used to characterize a fuel's chemistry and behavior, not necessarily as specific safety thresholds. However, a higher flash point is used to distinguish a greater safety factor in storage and fuel delivery. The US Navy uses JP-5 for shipborne aviation because it has a flash point of around 62C.
Neither of these temperatures is related to the auto-ignition temperature, at which the fuel will spontaneously ignite and burn. This is generally much higher than the fire point.