If someone is up-and-coming, they're verging on being really successful or famous. An up-and-coming rock band is just starting to be popular and beginning to draw crowds at their shows.
If you're an up-and-coming magician, you might be a little bit new to doing magic tricks, but you work hard at it and are actually pretty good. Local fans might be starting to mention your name when someone's looking for a birthday party entertainer: "Have you seen that amazing up-and-coming magician who pulls rabbits out of her hat and then turns them into bouquets of flowers?"