The Elder Scrolls V: Skyrim/A Daedra's Best Friend

Introduction
You are introduced to this quest in Falkreath when at least level 10, when a guard approaches you and asks if you seen a dog. If you ask for more information, the guard directs you to the blacksmith Lod. The blacksmith says he was looking for a dog that was wandering on the road, and gives you some meat to attract him.

When you head to the location, the dog introduces himself as Barbas, allows you to commence this quest.

If Lod dies before he tells you about the dog, you cannot start this quest.

Travel with Barbas to the shrine of Clavicus Vile
Clavicus Vile is locataed in the Haemar's Shame, and he will move ahead of you during this segment. As Barbas is invulnerable, you don't need to worry about his death, but will need to keep up with him.

When you reach the cave, you will need to fight vampires. The largest amount will be at the shrine proper, although you can attack isolated targets.

Speak to Clavicus Vile, and he will tell you to retrive the Rueful axe.

Retrieve the Rueful Axe with Barbas
When you reach Barbas, he will first be reluctant to reunite with Barba, but will eventually give you a quest to retrieve the Rueful axe.

Clavicus will act as a companion during this quest segment, and will follow you.

Return to the shrine of Clavicus Vile with Barbas and the Rueful Axe
If you are playing for the Oblivion Walker achievement but want to keep the axe, wait until you collected all other artifacts before completing this quest, and save your game. Once you receive the achievement, you may reload and choose the other option.

You may also delay this part of the quest if you want to keep Barbas for longer.

Return to Clavicus.

Here, you will have a choice. You can kill Barbas and keep the axe. This two-handed axe swings inflicts stamina damage, and has a base attack of 22.

If you prefer to get the Daedric artifact, insist that Clavicus keeps the axe. In return, you will get the Masque of Clavicus Vile, a helmet that provides base armor of 23, as well as improving speech (10 points), prices (20%), and magicka regeneration (5%).

Exploit: Keep both the mask and axe
A known exploit to keep both items exists. The exploit may need a few attempts, and is finicky. You need the Slow Time shout for it to work. It works best with the Whirerun home.

Equip the Axe, Slow Time shout, and position yourself behind the shrine to Clavicus, behind the chest. Make sure the exit is open. Save the game, and activate the shrine. As soon as you are able, use the shout, backup, and exit the cave. Fast travel to Whiterun (or wherver you have a home), enter it, and place the weapon on the weapon rack.

Once you receive the message that the quest is completed, return to the shrine to receive the masque.