Everyone wants an Super Dog! To do this you need to travel to Falkreath and find Lod, how is a smith. He will ask you to find his dog as part of a quest. The easy part is that his dog is outside the village (named Barbas).

Barbas will take you to the Shrine of Clavicus Vile and you need to speak to the Daedra Lord there and take on the "A Daedra's Best Friend" quest.

The dog will stay with you for the quest and since it is technically considered a quest item, it cannot be killed! It will also automatically attack all bad guys that come near you.

So just don't finish the quest and keep travelling around with your invincible dog!