There are many different ways to calculate how much two items are associated. The calculation of these correlation coefficients depend on how the items are measured. That is, the calculation depends in part on whether the items are measured on a nominal, ordinal, interval, or ratio scale.
The most common correlation coefficient is the Pearson r, which is named after Karl Pearson who developed it about 1900 (although some say that Galton should get credit for its creation). It is used to calculate the correlation between two continuous variables (ratio or interval scale).
Spearman ρ (rho)
The Spearman ρ is used to calculate a correlation for two variables that are ordinal (rankings). The greek letter ρ is used, which is an exception to the general rule that Greek letters are used for population parameters. Kendall τ (tau) is often used in place of Spearman ρ
The biserial correlation coefficient is used when one variable is continuous (ratio or interval scale) and the other variable is dichotomous (nominal). However, the dichotomous variable is thought to reflect an underlying continuous variable. For example, if we had high and low anxiety, we would expect that anxiety is actually a continuous variable. A similar correlation coefficient, called a polychoric correlation is often used.
The point-biserial r is used when one variable is continuous (ratio or interval scale) and the other variable is dichotomous (nominal scale). An example might be depression and gender.
Rank-Biserial Correlation Coefficient
The rank-biserial correlation coefficient is used to determine the association between dichotomous (nominal) and ordinal (rankings) data
ϕ (phi) Coefficient
If both variables instead are dichotomous (nominal), the ϕ (phi) correlation coefficient is used. The greek letter ϕ is another exception to the general rule that Greek letters are used for population parameters. A tetrachoric correlation coefficient is often used in place of ϕ.