Location: East-Southeast from Bruma (link goes to the Daedric Shrines Map at UESP Wiki)
Minimum Level: 5
Offering: Personality less than 20
Reward: Ring of Namira
Objective: Eliminate the Priests of Arkay at Anga
Another shrine that no one tells you about. Head almost due east from Bruma's south wall toward the "B" in "County Bruma" on your overland map. Hrolfrodi the Harrier, the Nord at the shrine, tells you that you're entirely too pretty and that people like you too much. You'll need to make yourself more loathsome before you can talk to Namira. This seems a bit difficult to do since potions, spells and items that damage attributes can't be used on yourself. However all is not lost. You do have a few options.
Namira wants you to get rid of the Priests of Arkay at the Ayleid ruin, Anga, a ways south of the shrine. Well, that's not quite true. Namira wants you to fix it so that the Forgotton Ones in the ruins can kill the priests. The problem being that the priests carry torches and the Forgotten Ones are afraid of the light. So your job is to douse their torches and let the Forgotten Ones deal with the priests. Since you probably don't have a spell or item that will do this, Namira provides you with "Namira's Shroud", a Novice-level Alteration spell, that puts out the torch (actually it just causes the priests to put their torches away - you'll find them in the priests' inventory).
There are four priests in the ruins, so just zap them and let the Forgotton Ones take care of the rest. If you're feeling particularly charitable, you can heal up any wounded Forgotton Ones after they take care of the priests, but it's not strictly necessary for the completion of the quest.
In the southern (well, southwestern) part of the ruins you'll find an altar for Namira with a button on the wall next to it. Push the button to open up a hidden room that has a couple of undeads and Ayleid containers. There are also a few containers and coffins scattered throughout the ruins, but there isn't anything to fight in the ruins aside from the two undeads.
Return to the shrine to collect your reward, the Ring of Namira. It's got a Reflect Damage and Reflect Spell effect on it that might be helpful.