An Expert Staff Ghost is a harder tier Staff Ghost that's unlocked by beating the normal Staff Ghosts within a certain amount of time. Unlocking a certain amount of Expert Staff Ghosts unlocks characters, Karts and Bikes, such as Funky Kong and the Sprinter. However they didn't return in Mario Kart 8, likely due to the slower pass of the game.
Expert Staff Ghosts make a return in Mario Kart 7. Unlike Mario Kart Wii, the game does not tell the player that they unlocked the Expert Staff Ghost, making it slightly harder to tell if the player unlocked the Expert Staff Ghost or not. But usually, the change in difficulty is a good way to tell.