When speaking in terms of flash specifications, what is flash duration and how does it impact exposure?
Simply, it's the duration the flash is actually on, emitting light.
This doesn't impact exposure per se; as long as the same amount of light is emitted, you'll get the same exposure. But it can affect the result: as the flash duration gets shorter, it has a better ability to freeze motion.
For most photography this won't matter very much. 1/1000s is a typical duration for decent flashguns at full power1, which is more than sufficient to freeze normal motion in a photograph. Other aspects will have much more bearing on the results; flash power, shutter speed, and so forth.
The one example I can imagine where shorter flash durations would be particularly desirable is for high-speed photography.
- Studio strobes are sometimes rated faster, but thanks to their different characteristics, may be slower in practice. More detail here: http://www.scantips.com/speed.html
Duration is how long the bulb is on, essentially, and so determines how much light is applied to the subject. The duration would then have an impact on freezing a subject, shorter durations able to freeze faster motion. This is very common in high speed photography such as water drops and the like where you apply a very short burst of light to freeze the drop and/or splash at that brief moment.
I use this technique a lot with water drops, my actual shutter is open for a second and I will apply a very short burst of light to freeze the motion. So, clearly, the room needs to be very dark, but now the shutter is irrelevant to stopping the motion, it's all in the light.
The duration is exactly that - the amount of time that the flash is lit for. The longer the flash is lit, then the more light can be used to illuminate the scene.
Flash power affects its "guide number" with more powerful flashes having a higher guide number and you can get the same amount of light out of a more powerful flash in a shorter time, which allows you to capture faster moving objects.
One thing I'd add to Matt's explanation would be that when you use high-speed flash sync (available on most same-brand DSLR+flash combinations) to get shorter shutter times below your camera sync speed (X-sync), the flash actually fires multiple times, so you can't use it to freeze fast moving subjects, because all you'd get would be several phases merged together.