Effect size is a measure of the strength of a phenomenon (for example, the change in an outcome after experimental intervention).

When you want to infer the relation ship in data, you can use the effect size. Effect sizes complement inferential statistics such as p-values.

On every occasion! Effect size fulfils an important requirement for the external validity of your research (the degree to which your research tells us something about the world in which we live). By translating the abstract language of numbers into real-life measures, we make the final step between question and answer.

Stating that there is a 'significant relationship' between Vioxx and risk of fatal and non-fatal heart attack is not sufficient. But when you estimate (as the FDA did) that Vioxx was responsible for around 27,000 fatal and nonfatal heart attacks you understand the extent of the manufacturers culpability in not withdrawing it when it was clear from the data that it was unsafe.

I completely agree with Ronán Conroy. You should always calculate an effect size. Even if the sample is non-random. It is important to know how large your effect is and that you are not simply overwhelming your statistics with a large N.

I also agree with Conroy and Rafael Gracia. To be more specific I will elaborate further.

In case of physics experiments dimensions M.L and T are used. When all are variables then size of each is important to obtain reproducible result. If only one is important say L then its size plays a crucial role. Say, in case of dispersed system particle size plays important role in deciding property of the system. Similarly in vibrating system frequency (time) plays prominent role.

In engineering and process equipments 'scaling' should be considered, that is from laboratory scale to manufacturing one should be aware of size of scaling.

In statistics one should note the sample size.

You can extend this argument in all the cases of data analysis.