How much does a machine learning algorithm leak about its training data, and why? Membership inference attacks are used as an auditing tool to quantify this leakage. In this paper, we present a comprehensive \textit{hypothesis testing framework} that enables us not only to formally express the prior work in a consistent way, but also to design new membership inference attacks that use reference models to achieve a significantly higher power (true positive rate) for any (false positive rate) error. More importantly, we explain \textit{why} different attacks perform differently. We present a template for indistinguishability games, and provide an interpretation of attack success rate across different instances of the game. We discuss various uncertainties of attackers that arise from the formulation of the problem, and show how our approach tries to minimize the attack uncertainty to the one bit secret about the presence or absence of a data point in the training set. We perform a \textit{differential analysis} between all types of attacks, explain the gap between them, and show what causes data points to be vulnerable to an attack (as the reasons vary due to different granularities of memorization, from overfitting to conditional memorization). Our auditing framework is openly accessible as part of the \textit{Privacy Meter} software tool.