Aspirations are your PLAYERS aspirations for directions they would like the story to go in. They tell the Storyteller what aspects of the story to focus on, and give the players a mechanical reward for acting towards those directions, by giving beats. Aspirations can be short-term (ex. Meet the Prince), or long-term (ex. Become the Prince). Short-term Aspirations give Beats when they're finished, long-term when significant progress is made towards it. Aspirations can also be negative for the characters (ex. Lose someone close to me), or mechanical in nature (Learn a new supernatural power), but I would consider those types of Aspirations as to be exceptional. Most aspirations you'll see will also be the Character's goals. Negative ones require a larger player-character disconnect than most RPG players are used to, and mechanical ones are just tricky.

For the ST, a positive story Aspirations gives them low-key feedback as to what type of story each player wants, and allows them to help the players advance their characters as they want. Negative story Aspirations tell the ST that the players won't get mad if the ST starts to REALLY bring the heat on them. And mechanical Aspirations give an opportunity for the ST to make a small story out of gaining a new Dot in something.