The Dog is one of the two pets you can choose in Stardew Valley, the other being a cat.

You can pick a name for your dog. He stays inside on rainy days, a good early indicator of the weather if you did not check the forecast the previous day. If the weather is good he can be found wandering your farm, mainly around his bowl, just North-West of your house. Other than that they're not all that helpful. Though you can pet and feed the dog, it is not necessary.