Logic is the study of correct arguments. An argument is anything that's offered to derive a desired conclusion from a collection of premises. The premises are meant to be truths that justify drawing the conclusion.
Logic can either be presented informally or formally.
Informal logic is what you might get in an introductory philosophy class. It's about being able to dissect an argument and find it's flaws or become convinced by it. The boundaries of informal logic are not always clear. It's making sure that premises properly justify a conclusion, but perhaps also investigating whether the premises themselves are true.
Formal logic is a mathematical theory which is meant to provide a model of one particular aspect of correct reasoning. An argument can be translated into formal logic by abstracting away from the content of the sentences in the argument to determine whether the form of the argument is correct.