Pearson product-moment correlation coefficient is measure of linear correlation and it is loosely related to variable dependence. Formula for calculating correlation coefficient of random variables `X` and `Y`is
where `mu` is expected value and `sigma` is standard deviation. ` `
To understand why this is called product moment we first need to know what is moment in probability theory. The `n`-th central moment of random variable `X` is `E[(X-mu)^n]`
So if we look at numerator of formula for correlation coefficient we see that it resembles second central moment (variation), but it's not exactly variation but rather moment of a product of two different random variables. Also in the denominator we have product of standard deviations (standard deviation is square root of second moment), hence product of moments. So correlation coefficient is (loosely speaking) moment of product divided by product of moments.
The name is also used to specify the type of correlation coefficient because there are other correlation coefficients e.g. rank correlation coefficients.