Someone who's destined is fated or meant to do something. If your mom says you're destined for a life on the stage, she means that it's certain you'll be an actor one day.
Being destined for something can be great luck — if, for example, you're destined to be a millionaire. It can also mean something closer to "doomed," as you might be if you're destined to a string of bad luck after breaking a mirror and seeing a black cat run past at the exact same moment. The Old French root is destinée, "purpose, intent, fate, or destiny."