In addition to their regular level, Captains also have a star level, which grants them additional bonuses.
The bonuses are permanent, assuming the relevant Captain is active.
You can see your current star bonuses in the Captain information window. You can view the bonuses provided by other star levels by hovering your mouse over the star.
Each star also provides you with faction points. The number of faction points that you can see on each star level represents the total number of faction points that you will end up with, including all previous stars.
To increase a Captain's star level, click on the arrow to the right of the stars.
You will see the required number of Captain fragments and Essences of the Immortals that you'll need in order to upgrade the star level.