The guide number is a value indicating the strength of a flashlight. According to Andreas Feininger the guide number helps finding the right aperture for flashlight exposures with the formula
aperture= guide number
distance between flash and image subject (in meters)
Since different manufacturers give guide numbers based on different filmspeeds (50 or 100 ASA) or even no guide number for their flashes Feininger suggests the user to find out the guide number himself by making 3 test exposures with the film he wants to use for flash photography. The image subject must have the same distance for each test exposure, distance and aperture must be notified. The aperture for the first test photo must be selected according to the formula when calculated with the official guide number. If the film has another speed than the ASA speed on which the official guide number is based this guide number must be recalculated for the other film speed. The further exposures shall be made with aperture below or above the value calculated for the guide number.
The best of the three pictures will give the guide number through the following formulas (based on distances in meters, speeds in ASA)
test guide number = official guide number * (√2) log2 ( film speed / speed of official guide number )
real guide number = aperture * distance between flash and image subject
The effective guide number can be different depending on factors like the subject's light reflection or alternate flash usage ways like indirect flashing or flash reflector adjustment.
Modern all-automatic cameras don't let the users worry about all that stuff. They optimize flash exposures with their through-the-lens metering "on the fly". But even for those cameras the guide number helps to understand the limited distance on which a flash is effective, see formulas
maximum distance= guide number