At work
When you see a flash working, you are probably left with the impression that it emits the light in a binary fashion: it is dark, then very bright and then dark again. So from zero to full speed in no time.
Actually your flash takes a little time to build the light intensity to peak, and also a little time to turn it off again, so you get a curve that looks like a wave. Flash duration is about how long time it takes from the flash is fired until the light is turned fully off again.
Measurement
If the flash emitted the light in a fully binary fashion, it would be easy to measure the duration of the light being on. But when you have a curve, you need to find a point on that curve to use. So typically the flash duration is measured from when the light has reach 50% of full throttle to when the light has either dropped to 50% of full throttle. An alternative is to do the same, but for 10% instead. These two measurements are often named t0.5 or t0.1.
For producers of flash, the competition to have the best numbers in the spec sheet often leads them to give the 50% value, as it is the fastest number. If they don’t tell you how they have measured, my bet is that they have taken the fast number and put in the spec sheet, i.e. t0.5.
Power
Another complicating factor is that the more power the flash lets out, the slower it typically gets. So the flash duration is very fast if you only use a fraction of the power the flash can provide. For this reason, the flash duration is often listed in the spec sheet as an interval. The Godox V860II has a flash duration listed as 1/300 to 1/20000 seconds.
And so what?
I Granted, there are probably many flash photographers out there that really don’t care too much about the flash duration, unless they are specialized in super high speed photography. For me the importance of flash duration is to realize that your flash (often) is much faster than your shutter speed, and this is vital in order to understand other flash related concepts like front curtain synchronization, rear curtain synchronization and high-speed synchronization.