Method of moments is thought to be one of the oldest, if not the oldest method for finding point estimators.  First introduced in 1887 by Chebychev in his proof on the Central Limit Theorem, the method of moments was then developed in the last 1800s by Karl Pearson.  In 1936, he published a paper that was highly critical of a colleague of Ronald Fisher, who responded strongly in a paper in 1937.  The 1936 and 1937 papers are easily found online.  

Essentially the method of moments equates the first k sample moments with the first k population moments, resulting in a system of k equations and k unknowns, where the unknowns are the parameters.  The system is then solved for the parameters, yielding estimators for the parameters in terms of the sample moments.