Once you found a wild dog you can feed it dog biscuits to tame it. After you have tamed your dog it will not follow you until you right click it with a dog collar and then right click it with a leash. To make it stop following you right click on your dog while holding the leash and the dog will free roam.
If your tamed dog gets injured you can not feed it directly. Placing a dog dish down and right clicking on it while holding a meat will fill the dish and your dog will eventually go over and eat to regain health.
Put two tamed dogs near a torch or fire and feed them both a cookie. They will immediately produce a puppy that look like miniature versions of their parents.
Dog Biscuit: Dog Leash: Right click item on a tamed dog wearing a collar.
Dog Collar: Right click item on any tamed dog.
Dog Dish: Right click on the dish while holding meat to fill.