Classical: The standard kinds of logic that you learn in undergraduate logic classes are the best (or right) logics, the ones that best model (ETA: idealized versions of) our inferential processes. Examples of classical logics are Boolean logic and first-order predicate calculus. Classical logics are bivalent (sentences can only be true or false), obey the principle of the excluded middle (if a proposition is not true, its negation must be true) and obey the law of non-contradiction (a proposition and its negation cannot both be true).

Non-classical: The best logic is not classical. Non-classical logics usually reject the principle of the excluded middle or the law of non-contradiction. An example of a non-classical logic is dialetheism, according to which there are true contradictions (i.e. some sentences of the form "A and not A" are true). Proponents of non-classical logics argue that many of our scientific theories, if you probe deeply, involve inconsistencies, yet we don't regard them as trivially false. So they claim that we need to revise the way we understand logic to accurately model our inferential processes.