Propositional logic, also called $0$th-order logic and sentential logic, is that part of logic that deals only with propositions with no bound variables.

Compare predicate logic, or $1$st-order logic, and higher-order logic. Note that while one can have free variables in $0$th-order logic, one cannot really do anything with them; each $P(x)$ in a $0$th-order proposition might as well be thought of as atomic.

This can be understood more cleanly in the language of many-sorted logic, where each variable has to have a specified sort. Then ordinary predicate logic has exactly one sort, usually unnamed. Propositional logic is for a signature with no sorts, hence no variables at all.

A propositional calculus, also called sentential calculus, is simply a system for describing and working with propositional logic. The precise form of such a calculus (and hence of the logic itself) depends on whether one is using classical logic, intuitionistic logic, linear logic, etc; see those articles for details.