There are a couple of dogs that you can get in Skyrim as a companion. More are added in the Dawnguard DLC, but the best dog available is Vigilance, which can be found in the basic Skyrim game.
Finding Meeko.
Meeko is located at Meeko's Shack out in the woods, east of the Dragon Bridge and south of Solitude Sawmill. When you first find the shack, it's possible that Meeko will not be located inside of it. If this happens walk around outside of the shack and he may be found outside wandering. If you manage to find him he will run back to the shack to whine at the loss of his dead owner. Talking to him here will allow you to ask if he wants to come with you, and from then on he can be your companion. If you can't find him at all, sometimes fast traveling away and then back will have him appear inside of the shack as he is supposed to.
Finding Bran the Husky.
For this dog you'll need to have the Dawnguard DLC and access to the Dawnguard fortress. Bran is located inside by a small campfire, and talking to him you can invite him to come with you to hunt some vampires.
Finding Vigilence, the war dog.
This dog has the best stats of all the others, and will make a good companion in a fight. He doesn't come freely though, you'll have to buy him. Head to the Markarth stables and talk to Banning here. He will offer to sell you the dog for 500 gold.
There is also a random encounter that you can run into out in the world where you can find a stray dog being attacked by two wolves, if you manage to run into that you can kill the wolves in time to save the dog, and have him follow you around as well.
Finding Meeko.
Meeko is located at Meeko's Shack out in the woods, east of the Dragon Bridge and south of Solitude Sawmill. When you first find the shack, it's possible that Meeko will not be located inside of it. If this happens walk around outside of the shack and he may be found outside wandering. If you manage to find him he will run back to the shack to whine at the loss of his dead owner. Talking to him here will allow you to ask if he wants to come with you, and from then on he can be your companion. If you can't find him at all, sometimes fast traveling away and then back will have him appear inside of the shack as he is supposed to.
Finding Bran the Husky.
For this dog you'll need to have the Dawnguard DLC and access to the Dawnguard fortress. Bran is located inside by a small campfire, and talking to him you can invite him to come with you to hunt some vampires.
Finding Vigilence, the war dog.
This dog has the best stats of all the others, and will make a good companion in a fight. He doesn't come freely though, you'll have to buy him. Head to the Markarth stables and talk to Banning here. He will offer to sell you the dog for 500 gold.
There is also a random encounter that you can run into out in the world where you can find a stray dog being attacked by two wolves, if you manage to run into that you can kill the wolves in time to save the dog, and have him follow you around as well.