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An effect size is a standardized way to report the strength of an apparent relationship. For example, assume that you are evaluating a new treatment for OCD. The effect size of the treatment answers the following question: "how much does a typical patient benefit from the treatment?".
It is important to note the distinction between effect size and statistical significance. In particular, statistical significance indicates the likelihood that an observed phenomenon is real, regardless of the strength of the phenomenon. Therefore, it is possible to have a small effect size that is statistically significant. It is considered best practice is to always report an effect size with each statistically significant result.
The way effect size is measured depends on the statistical test being conducted. Most of the online tests on this site report an effect size.
Instructions  

Difference between two means  If you have the means and standard deviations of the two data sets, use the Cohen's d calculator at the bottom of this page. 
Correlation  The correlation coefficient is reported on the correlation page. 
Regression  rsquared is reported on the regression page. 
Independent ttest 

Paired ttest 

Wilcoxon ranksum test 

MannWhitney test 

Wilcoxon signedrank test 

Oneway ANOVA 

Repeated measures ANOVA 

* Effect sizes are computed using the methods outlined in the paper "Olejnik, S. & Algina, J. 2003. Generalized Eta and Omega Squared Statistics: Measures of Effect Size for Some Common Research Designs Psychological Methods. 8:(4)434447".
If you are comparing two populations, Cohen's d can be used to compute the effect size of the difference between the two population means. By convention, the Cohen's d is categorized as follows:
Cohen's d  Interpretation 

0.2  A small effect 
0.5  A medium effect 
0.8 +  A large effect 
The following calculator computes Cohen's d using the mean and standard deviation of the two samples: