To outlast is to live or survive longer than someone (or something) else. Sturdy old buildings will outlast newer, more flimsy and cheaply made structures.
Typically, your parents will outlast your grandparents, and you will outlast your parents. You might convince yourself to buy some outrageously expensive shoes by thinking about the fact that they're so well made, they'll probably outlast the much cheaper pair you were considering. And you might wonder why a terrible TV show has been on for years, managing to outlast several brilliant ones that were cancelled after a single season.