Many of these missions require that you receive a specific score in a minigame. In order to find out what score you need, just do the minigame once and then check the journal. The score needed will be listed under that minigame.
This section of the journal includes, but is not limited to:
Replaying minigames that you did during the main story, such as the magic carpet escape sequence in Agrabah. Generally you can go to the area where you did the minigame originally and there will be an NPC you can speak to and replay it.
Replaying each song with the Little Mermaid and replaying each minigame with Winnie the Pooh and getting a high score.
Defeating every tournament in the Underworld Arena and getting a good score. There are 8 total.
Replaying all of the money making methods that were available during the Roxas section of the game in Twilight Town. Each of these also has a high score that needs to be beaten.
Beating each opponent in Struggle in Twilight Town. You must defeat each opponent ten times before the next will become available, there are three opponents altogether. Once you are in possession of 200 orbs the game will end immediately, so if you are a higher level you can get ten wins quickly.
Each member of Mushroom XIII needs to be beaten with a certain score or under a certain time. There are 12 challenges you must win, then go to where the "Challenge of the 1,000 heartless" took place to unlock the 13th and final Mushroom XIII.
Defeating Sephiroth, The Lingering Will, each replica Organization XIII member, and each Absent Silhouette.
Getting certain scores while riding the skateboard in some worlds. There will be an NPC you need to speak to before you can start earning points.