Covenant Renown is a main Covenant progression system required to improve your trust with a Covenant and unlock unique Covenant Rewards.
There are 2 primary ways to gain Renown with your covenant: completing the Covenant Sanctum Weekly Quests and progressing through the Covenant Campaign chapters.
Sanctum Weekly Quests are 2 weekly quests offered by your Covenant in your Covenant Sanctum: Replenish the Reservoir and Return Lost Souls. The former will require you to gather Anima for your Sanctum deposit while the latter will require you to rescue Tormented Souls from the Maw. These 2 weekly quests are the primary ways of raising Renown Levels with your Covenant.
The Covenant Campaign is the main questline that dives deeper into the plot of your Covenant and reveals more of their story and goals. It also unlocks unique Covenant rewards.