The extension of an idea consists of the things that it applies to; it contrasts with intension. This general notion is from semantics, and has been applied to some other fields:
In philosophical semantics or philosophy of language, the extension of a word, phrase, or concept is the set of things it extends to, or applies to. So the extension of the word "dog" includes all the dogs in the world: Fido, Rover, Lassie, Rex, and so on. The extension of the phrase "Wikipedia readers" includes each person who has ever read Wikipedia, including the person reading these words right now. Extension is one of the meanings of "meaning". Many philosophers argue that the meaning of a word is sometimes just the things that the word picks out, or applies to.
Sometimes additional wrinkles are thrown in. For example, if "dog" referred only to all the dogs currently in the world, then it would be technically incorrect to refer to an imaginary dog or a dead dog as a "dog". Since this is an unsavory conclusion, the extension is often taken to pick out objects not in the current world but in a set of "possible worlds". In this case, the extension of "dog" would include all the dogs in all possible worlds, including worlds consistent with the past, or imaginary worlds.
In mathematics, the extension of a mathematical concept is the set that specifies that concept. This can mean different things in different cases, and there is no universal definition of the term "extension". For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function; in other words, the function's graph. The extension of an object in abstract algebra, such as a group, is the underlying set of the object.The extension of a set is, of course, the set itself. That a set can capture the notion of the extension of anything is the idea behind the axiom of extensionality in axiomatic set theory.