A class usually only offered at college that deals with the analysis and appraisal of arguments. There are different systems of logic for different uses, including syllogistic logic (an ancient form of logic dealing with categories), propositional logic (dealing with statements dependence on each other), quantificational logic (a sort of combination of syllogistic and propositional), modal logic (dealing with necessity and possibility), deontic logic and imperative logic (dealing with "can," "should," "will," etc.) and belief logic (dealing with patterns of belief). Most of these systems can be combined with each other and I'm pretty sure there's Wikipedia articles about all of them if you want to know more. Some classes will also cover formalized ethics (building an ethical theory with these previous systems), metalogic (the study of the logical systems themselves) and various forms of reasoning and argumentation.
It is sometimes described as math without numbers, because you learn various formulas to calculate whether or not a statement is true or false in a particular context, but the only numbers you ever see are 0 for false and 1 for true. It is also possibly the worst college course about having people bitch about how hard it is when in reality it's one of the easiest courses you can take.
Imagine a kid with Down's Syndrome handing you a piece of paper with various statements relating to the same thing on it and asking you if they make sense together. If you could handle this, you can handle a logic course. If not, you probably need to take a logic course.