A watt second (also watt-second, symbol W s or W·s) is a derived unit of energy equivalent to the joule.(Taylor & Thompson 2008, pp. 39–40, 53) The watt-second is the energy equivalent to the power of one watt sustained for one second. While the watt-second is equivalent to the joule in both units and meaning, there are some contexts in which the watt-second is used instead of joule.
In photography, the unit for flashes is the watt-second. A flash can be rated in watt-seconds (e.g., 300 Ws) or in joules (different names for the same thing), but historically the watt-second has been used and continues to be used. To know the watt-seconds of a flash, you need to know two things: the voltage that the capacitor/flash tube are working at, and the value of the storage capacitor connected to the flash tube. With those two values, the formula is half the value, in Farads, of the capacitor, times the square of the voltage. For example, an on-camera flash, using a 1000 microfarad capacitor at 300 volts, would thus be 45 watt seconds. Studio flashes, using larger capacitors and higher voltages, are in the 200 - 2000 watt-second range.
The energy rating a flash is given is more of a technicality than a true benchmark in terms of light output because there are numerous factors that affect the energy conversion efficiency. For example, the actual construction of the tube will vary the efficiency but also the use of reflectors and filters will change the usable light output at the subject. Some companies specify their products in "true" watt-seconds and some specify their products in "nominal" watt-seconds. 
Companies that use watt-seconds to describe their products:
- Guide number
- Kilowatt hour, a scaled unit of the watt-second often used in the electrical energy industry
Taylor, Barry; Thompson, Ambler (2008), The International System of Units (Special publication 330), Washington: National Institute of Standards and Technology