Weighted voting systems play a crucial role in the investigation and modeling of many engineering structures and political and socio-economic phenomena. There is an urgent need to describe these systems in a simplified powerful mathematical way that can be generalized to systems of any size. An elegant description of voting systems is presented in terms of threshold Boolean functions. This description benefits considerably of the wealth of information about these functions, and of the potpourri of algebraic and map techniques for handling them. The paper demonstrates that the prime implicants of the system threshold function are its Minimal Winning Coalitions (MWC). The paper discusses the Boolean derivative (Boolean difference) of the system threshold function with respect to each of its member components. The prime implicants of this Boolean difference can be used to deduce the winning coalitions (WC) in which the pertinent member cannot be dispensed with. Each of the minterms of this Boolean difference is a winning coalition in which this member plays a pivotal role, in the sense that the coalition ceases to be winning if the member defects from it. Hence, the number of these minterms is identified as the Banzhaf index of voting power. The concepts introduced are illustrated with detailed demonstrative examples that also exhibit some of the known paradoxes of voting- system theory. Finally, the paper stresses the utility of threshold Boolean functions in the understanding, study, analysis, and design of weighted voting systems whatever their size might be.