There are 6 basic types of missions, all of which will be explained below.
Story Missions
These missions are available at all times and are the standard source for Experience, Gold, and Units. They also each give a bonus silver unit when you have found all drops in one mission. The missions cost Charisma to play. SCs are provided for every first 3 star clear. These mission also have the option for a 4th star but it comes with gimmicks, stronger enemies and various restrictions on players depending on the mission.
Challenge Quests
These missions are available at all times and are the standard source for Spirits and Gifts. They are typically short and simple, but come with a special gimmick, such as ranged units only or fast enemies. They cost Stamina to play.
Daily Missions
These missions change each day and always follow the same pattern. They provide a wide variety of useful items for the player.
All Dailies except Awakening Missions cost Charisma and Stamina depending on their grade.
Event Missions
Event Missions are missions that appear for a limited time. These make up a large majority of new content for the game. Each involves a set of maps with their own special conditions and stories. The goal of most events is to complete said conditions, and to obtain the best possible copy of the event's unit (most commonly of platinum rarity).
Tutorial Missions
There is a tutorial and is always available: Katie's Strategy Lessons. In Katie's Tutorial you get an introduction to basic game mechanics.
Bonding Quests
Bonding Quests (Exchange Quests, 交流クエスト) are special missions available for certain Gold+ units after Awakening. New Bonding Quests are being added over time.
Only the subject unit, the Prince, and a pre-set team will be available for deployment during the missions.