When should I use each of these methods to calculate correlation? I am using R for data analysis. R provides a corr function for calculating the correlation. This function provides three different approaches/algorithms to estimating the corr which are Pearson, Spearman and Kendall. When should I use each of each of these methods? What factors determine which method should be used?
 A: Pearson's product-moment coefficient (pearson parameter) measures linear correlation between variables. Therefore it is appropriate when your suspected correlation is linear, which can be visually inspected with a plot. 
Kendall Tau coefficient (kendall paramter) and Spearman's correlation coefficient (spearman parameter) are measures rank correlations. So the correlation between the two variables does not need to be linear. spearman method is basically the pearson method, but applied on the ranks of the values (the rank of a value is given by it's position after sorting the values). kendal method is build basically as a statistic in a form of a ration between the additional number of ordered pairs and the total number of pairs. For kendal method, because it is build as a statistic, one can build also use it in the framework of hypothesis testing, with all the benefits (it is called tau test).
All these methods are instruments used to infer something about the dependencies between random variables. See more on Wikipedia dedicated page dedicated to Correlation and Dependence 
