The Fortune Arcana
In order to unlock the Fortune Arcana in Persona 3 you will need to join one of the three culture club after school. Your options are the Music Club, Art Club or Photography Club. The Fortune Arcana is tied to a senior by the name of Keisuke Hiraga. On top of unlocking the Fortune Arcana from joining a club you'll also open up the possibility of raising the Priestess Arcana which can only be started once you've joined a club and grown closer to Fuuka. More information for that Arcana can be found in the associated guide.
Keisuke is available to you on June 17th and later on Tuesdays, Wednesdays and Thursdays. He can be found in the same hallway as your classroom on the second floor so when he is available to hang out he's pretty hard to miss.
While raising a Social Link you'll be given story cutscenes where you'll have to answer questions. The answers to each of these questions is different depending on the person and the situation. What you usually need to try and do is pick an answer that reflects their personality. Here's a cheat sheet for each Social Link rank of the Fortune Arcana and what to say.
Rank 1: Automatically earned
Rank 2: Anything - anything
Rank 3: Great job - anything
Rank 4: Anything - Complaining wont help
Rank 5: You should tell your father
Rank 6: Anything - Do whatever you want
Rank 7: Are you going to be a doctor?
Rank 8: Yeah I'm fine - Anything - Anything
Rank 9: (Try to Stop him) - Anything - Anything - Anything - Anything
Rank 10: Automatically earned
Upon completing the Fortune Social Link you'll be able to fuse the Persona Norn who has access to the unique spell Panta Rhei. You can fuse Norn through triangle fusion using Clotho, Lachesis and Atrophos.