Assignments are specific tasks the Avengers can complete alongside any mission to earn Faction experience. They can be found on a tab under Objectives in the JARVIS menu in-game.
Different Factions each have their own in-game Coordinator at their respective Outpost who give daily and weekly assignments that help increase your level with their faction. The higher your faction level, the more Gear and Artifacts the Faction vendors will give you access to. Faction items rotate daily and weekly. Make sure to go grab all your Faction dailies when you login, because they can be completed as you play through other missions.
Conquer: complete a War Zone
Harvest: collect the given Resource from various sources
Honor: gain experience
Showcase: perform special moves, or rescue allies
Terminate: defeat specific enemy types
Toil: open strongboxes
Triumph: complete War Zones on Challenge Rating III or higher
Vanquish: defeat enemies in a specific way