If a period of time elapses, that time goes by. Three days elapsed means three days passed. The subject is almost always a length of time — seconds, weeks, years — not a person and not a thing you hold. You cannot say I elapsed two hours; you spent them, or they elapsed. It is somewhat formal and more at home in writing, news, and exam English than in chat. In speech people usually say pass or go by. You often meet it when nothing much happens on the surface, but the clock still moves: Several months elapsed before the results came back. It does not mean to slip out of a room or to make a mistake; those are other cousins of the old Latin idea of slipping.