A sort of semi-awake train of thought, which is Never Remembered.
Interrupted, usually, by an alarm clock. The only sign that it happened is a sort of "What was I thinking about?!" kind of feeling, similar to when one is lost in thought, and suddenly, preferably loudly, violently, or embarassingly interrupted. Extremely annoying. Not to be confused with the feeling that you were dreaming and forgot what happened, as in either case you can normally remember a hint of something.
Dreams are normally much more exciting.