The Dog is one of the two pets you can choose in Stardew Valley, with the other option being the cat. At the Character Creation menu, you can choose between 3 different dog options.
Once Marnie shows up on the 10th day, you may pick a name for your dog. It 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 it can be found wandering your farm, mainly around its bowl, just North-West of your house.
You can pet it and fill its water bowl daily, and eventually you will get the two-heart message “[Pet] loves you”, which counts towards one of the points at Grandpa’s Shrine.