A Mission is a task provided to the player to complete in order to earn a reward, usually a Capsule. Reaching a new Rank requires completing a certain number of missions. Each mission completed will reward a Wood Capsule or Concrete Capsule, and ranking up will reward a Rank Up Capsule. Some missions also reward Gems or XP.
Most ranks have three more missions than you need to rank up. These are optional, bonus missions that may be completed for extra Capsules or skipped depending on your preference. These extra missions may also give you options if you want to skip a particularly troublesome mission encountered earlier in the rank (eg. Level Up Heroes when all your Heroes cost 20k+ XP to upgrade).