1. A general state, usually used with an adjective describing the state. Use of the word "mode" rather than "state" implies that the state is extended over time, and probably also that some activity characteristic of that state is being carried out. "No time to hack; I'm in thesis mode."
In its jargon sense, "mode" is most often attributed to people, though it is sometimes applied to programs and inanimate objects. In particular, see hack mode, day mode, night mode, demo mode, fireworks mode, and yoyo mode; also chat.
2. More technically, a mode is a special state that certain user interfaces must pass into in order to perform certain functions. For example, in order to insert characters into a document in the Unix editor "vi", one must type the "i" key, which invokes the "Insert" command. The effect of this command is to put vi into "insert mode", in which typing the "i" key has a quite different effect (to wit, it inserts an "i" into the document). One must then hit another special key, "ESC", in order to leave "insert mode". Nowadays, modeful interfaces are generally considered losing but survive in quite a few widely used tools built in less enlightened times.
3. <hardware> video mode.