Something that's elapsed has passed or gone by. Even five elapsed minutes can feel like forever when you're waiting to hear good news.
If your grandmother speaks wistfully about elapsed years, she's thinking about all the time that's gone by in what seems like an instant. When you're taking an important exam, you might keep an eye on the clock to keep track of the elapsed minutes. The word comes from a Latin root, elapsus, "slipped or glided away," combining ex-, "away," and labi, "to slip or glide."