A Mission is like a quest, but for a Garrison follower. The follower will probably get some form of experience, but the player gets the reward!
You'll send your followers on missions for varied purposes. Successful missions will give you the resources you need to keep developing your Garrison, but you'll also have the chance to acquire powerful loot for your character, and your followers will gain experience for each mission they run.
Each follower, depending on race, class and spec, have different abilities and scenarios that are best suited for them, such as a character who can taunt or a character who works better in a group.
The length of mission depends on the type of mission chosen, ranging from a few hours to a week. Missions can also be run while the player is offline.
The resources you'll acquire during missions include both existing crafting reagents and Garrison-specific materials. For example, if you assign a follower to a mining mission, you could receive ore, but you might also unearth stone, a new kind of resource used to construct and upgrade buildings.
Though followers are great for solo play, the system also has multiplayer elements. When you dispatch a party of your followers on certain missions, you'll also be able to join forces with your guildmates' followers, even if your guildmate is offline.