A proposition is a statement which is either true or false. In propositional logic, we reason only about propositions and see what we can do with them. Since this is mathematics, we need to be able to talk about propositions without saying which particular propositions we are talking about, so we use symbolic names to represent them. We will always use lowercase letters such as p, q, and r to represent propositions. A letter used in this way is called a propositional variable. Remember that when I say something like “Let p be a proposition”, I mean “For the rest of this discussion, let the symbol p stand for some particular statement, which is either true or false (although I am not at the moment making any assumption about which it is).” The discussion has mathematical generality in that p can represent any statement, and the discussion will be valid no matter which statement it represents.
Propositional variables are a little bit like variables in a programming language like Java. A basic Java variable such as int x can take any integer value. I say there is “a little bit” of similarity between the two notions of variables—don’t take the analogy too far at this point in your learning!