Pearson product-moment correlation coefficient

Formula
$$r_{X,Y}={\mathrm{cov}(X,Y) \over S_X S_Y}$$

Normality
The Pearson product-moment correlation is a parametric statistics and thus assume that the univariate distributions are approximately normal. Consider descriptive, inferential, and graphical indicators of normality - see Testing for normality.

Ultimately, if still in doubt, then conduct both parametric and non-parametric analyses of association between two variables and compare the results. If they indicate the same conclusion, then it really doesn't matter which technique is used. If the results are different, then the researcher should try to work out why and which is more appropriate. If still in doubt, then report both.