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There are seven Daedric shrine statues in the game that you can activate to receive quests. These quests result in the acquisition of various items. This is not a formal quest set, so there is no order in which you have to visit the shrines, nor is there anyone you have to speak to (other than the statues themselves – your quest-givers, as such). You'll want to mark your locations near the statues before setting off on these quests, since most of the statues will send you off to some remote location for their quest.
The Shrine of Azura is on the southeast corner of the mainland. The statue is you activate is inside the shrine beneath the giant landmark statue. When you click on the statue, you get a long-winded speech that boils down to this: go to the thick, fat island north of the island with Dagon Fel – it's straight north from the Pudai Egg Mine. On this island are various Daedra, most notably, a golden saint named Staada. Kill Staada and take Sheogorath's Signet Ring from her. DO NOT enter the shack nearby. Return to Azura's shrine with the ring and you'll receive Azura's Star – a re-useable soul gem.
Head to Hla Oad, and then head northwest to the Daedric shrine of Ashurnibi. Dive into the water west of the shrine and head down. You'll spot underwater Daedric Ruins, central to which is a broken statue whose head is on the ocean floor. The head is the part of the statue you want to talk to. The quest the head gives you will result in the creation of a new shrine to Boethiah located on Khartag Point. You'll need to go to Jobasha's Rare Books in Vivec to purchase a copy of Boethiah's Glory. Take that book and 2,000 gp to Duma gro-Lag, located in Ghorak Manor in Caldera. This Orc sculptor will create the new shrine. Once you've set him upon this task, you'll have to wait 20 days for his work to be complete. After the time has elapsed, visit Khartag Point and you'll see the new statue. Click on the new statue and you'll receive Goldbrand, a katana that does fire damage.
Note: if you become a vampire, and you're doing the quest to kill Shashev (in Ald'ruhn's Ald Skar Inn) for Sirilonwe (in the Vivec Mage Guild), you can get an upgraded version of Goldbrand. After killing Shashev, report back to Sirilonwe with Goldbrand and exactly 11,171 gp in your inventory. When you talk to Sirilonwe to complete the Shashev quest, Goldbrand will be replaced with Bluebrand, a differently colored and more powerful version of Goldbrand. You will also see the message, "Go to hell, Carolina!" Make of this what you will.
The shrine to Malacath is in Assurdirapal, which is on the eastern tip of the long island west of the Sanctus Shrine. This quest will direct you to Gnaar Mok. South of Gnaar Mok will be a dead bull netch and three people, one of who is Farvyn Oreyn. You have to kill Farvyn, and will probably have to kill his buddies as well. Once you've done this, Malacath will give you the Helm of Oreyn Bearclaw.
Note: the corpse of the dead netch will probably interfere with looting the bodies of your dead foes. The netch corpse cannot be disposed of. Solutions: lure your foes far away from the corpse before slaying them, or having slain them, open the control console and type in "TCL" to turn off the clipping – this will allow you to navigate beneath their corpses to loot them. Just remember to type in "TCL" again to turn the clipping back on afterward.
This shrine is in Yasammidan, which is west of Ald Velothi. The statue will send you to the Alas Ancestral Tomb, which is southwest of the Erabenimsun Camp. Inside the tomb, look for the skeletal remains of Varner Hleras. Take the rusty dagger from his corpse and return it to Mehrunes Dagon. He will replace the dagger with Mehrune's Razor, an enchanted dagger.
Find the Morag Tong headquarters inside the Vivec Arena Storage area. Inside, speak to Taros Dral about "sensitive matters" and he'll ask you to poison Balyn Omarel in Balmora. Taros will give you some treated bittergreen petals. Go to Balmora and enter Balyn's house from the roof without being seen by Balyn, who is outside on the ground level. Inside Balyn's house is a pot - click on it and choose to place the petals within. Go back to Taros and he will tell you to activate the square shrine behind him. Mephala will reward you with the Ring of the Khajiit.
Note: if Taros does not have a "sensitive matters" dialogue topic, then activate the command console, select Taros, and type in: AddTopic "Sensitive Matters". The topic will now be in his dialogues and you can commence with the quest.
This shrine is in Yansirramus, which is west of Tel Aruhn, on the largest island in the westernmost part of the bay. Molag Bal will set you upon a quest to Kora-Dur to kill a Daedra named Menta Na. Kora-Dur is east of Kogoruhn, north of the Ahanibi-Malmus Egg Mine, and southwest of Bthuand. Seek out Menta Na among the Daedra inside and report his demise to Molag Bal. In return, you will receive the Mace of Molag Bal.
In the Vivec St. Delyn Canton's Underworks is the Ihinipalit Shrine. The statue within will send you to the northernmost island on the map, the westernmost island of the three little islands up there. Once on the island, visit Big Head's Shack and talk to him. Among the nonsense he spews, he'll offer you the Fork of Horripilation on the table nearby. Take it and travel to the island to the east of the one you're on. There is a giant bull netch on that island, and you must kill it with the Fork. You can work around the odious chore of stabbing it to death with a fork by doing most of your damage with another weapon – but just make sure that it is the Fork that delivers the killing blow. Return to the Sheogorath statue and you will receive the Spear of Bitter Mercy.
And that is it for the Daedric shrine quests.