If you call someone illogical, they'll probably take it as an insult, but chances are they'll only have a vague idea of what logic is and how it works.
Logic is the branch of philosophy which studies reasoning.
There are many different types of logic, but we aren't going to concern ourselves with flow charts, boolean operators, and that sort of thing. Inductive logic is the more lenient form of logic. It can only be used to show that something is likely to be true or false. There will always be some amount of doubt about the conclusions made with inductive reasoning. Inductive logic is typically used in science to find general truths based on specific observations.