One of the quest categories that are available in The Outer Worlds are faction quests. They are commissioned by the leaders of the groups and their execution will increase your reputation with a specific faction. This page of our guide contains a list of faction quest.
Quests commissioned by the Iconoclasts are related to the main plot. Therefore, they must be completed.
Quests of MSI are also connected with the main story. Therefore, they must be completed.