What is a Watt Second?
Well, it's a unit of electrical energy, also known as a Joule. It is a Power of 1 Watt for a time of 1 second.
So what is a Watt? Well it is calculated as current (measured in amperes) multiplied by voltage (measured in volts).
So 1 Watt = 1 Ampere x 1 Volt = 0.5A x 2V = 2A x 0.5V etc. etc.
So, supposing you have a 1000W strobe, it has the capability to put 1000W of electrical power into firing the flash. This has no real bearing on how bright the flash is, although everything else being equal, a 500W strobe would, in theory, be half as powerful as a 1000W strobe.
Why Watt Seconds then?
A flash fires for only a fraction of a second. Depending on the make and model, this can be anything from 1/750s to 1/2000s or shorter. Generally, the harder a flash fires, the longer the flash time is. In order to compare the wattage with that of a standard continuous light source, such as a tungsten bulb, you really need to take into account how much power is used in a given time. So, taking a 500W tungsten lamp, you can propose that running it for 1 second will take 500 Watt Seconds, which I will say as 500Ws from now on. A 500Ws flash will use a far greater power, but does so for a much shorter time. Say the 500Ws flash fires for just 1/1000s, the actual power which would be used by the flash if it lit for the whole second would be 1000 times the 500Ws value, which would be 500,000W. That would take a seriously bright light to sustain that output!
So the answer is that Watt Seconds allow the electrical power used by a continuous lamp to be compared with that of a strobe.
How is light output measured?
If you buy a scientific light meter, you'll find that when you shine a light on it, it gives a value in Lux. This is the standard accepted unit for measuring the intensity of light which falls on a surface.
The output of a light source is measured in Lumens, which is defined by the following:
1 Lumen = 1 Lux x 1 Square metre
For the same reasons of comparing with continuous lights, it is necessary to make up a value known as Lumenseconds.
How are Watt Seconds and Lumenseconds linked?
The job of a strobe is to convert electrical energy into light energy, and like any light bulb, there is an associated efficiency. Some strobes are more efficient than others, with poor models producing around 15 lumenseconds for every watt second, while more efficient models can product up to 50 lumenseconds for every watt second.
So where does the term Effective Watt Seconds come from?
According to white-lightning.com, which gives the best explanation I have found, a company called Inverse Square Systems first coined this term in relation to a product known as "Stroblox" which it released in 1985.
They had produced a strobe which was more efficient than many others on the market. Most people compared one strobe to another in terms of Watt Seconds, because the efficiency of each strobe wasn't that different at that time. So in order to market this new product, they had to show that it had a lot higher light output due to higher efficiency than the equivalent watt second strobe from a competitor.
They therefore measured the average light output for competitors strobes, compared it with their own, and came up with the statement that their strobe was effectively as powerful as a competitors strobe with a much higher watt second rating.
It was a nice bit of marketing, but unfortunately the term has stuck, and despite being pretty much meaningless, successive companies happily quote an "effective watt seconds" rating, higher than the actual watt second rating of the unit, in order to make the product seem more powerful.
Quick conclusion
So how can you compare strobes? The only reliable way is to find data for the Lumenseconds output of the units, and compare all the units that way. If you have quoted Watt Seconds, you then need to know the efficiency in terms of the number of Lumens per Watt Second the unit produces. You can then calculate the Lumenseconds value to compare the units.
As for Effective Watt Seconds? It's very useful for selling underpowered strobes.