Those who own, harbor or keep a dog may be held responsible for the payment of compensation if it harms a person, a domestic animal, a farm animal, or property of any sort. Liability depends on whether there was negligence, a violation of an animal control law (such as a leash law), and particular provisions in the state, county and municipal law where the incident happened.
A dog owner can be held responsible for his negligence if he does something unreasonable with this dog or fails to take a reasonable precaution. For example, if you own a dog that has the tendency to attack other dogs, it would be unreasonable for you to bring that dog into a dog park. Therefore if your dog attacked another dog in a dog park, the owner of the other dog could hold you responsible for negligence.
Violating an animal control law pertaining to health or safety is always considered to be negligence. For example, if you are walking your dog without a leash, and your city or county has a leash law, you can be held responsible for anything bad that happens as a result of your dog being off the leash. If it chased a child who was riding a bicycle, and caused the child to have an accident by falling off the bike or crashing into something, you would be held liable for that.
Additionally, there are some places that have ordinances making dog owners responsible for any damage caused by their dog, regardless of how it happened. In these places, if you were walking your dog on a leash and it injured another person or dog in any manner, you would be held liable under the ordinance even if you were not negligent. Usually a dog owner will not be held responsible, however, if his dog is provoked or if somebody is hurt while trespassing on the dog owner's owned or rented property.
The foregoing sets forth general legal principles applicable throughout the United States. To learn the specific law applicable to a particular jurisdiction, one must read the the state laws, municipal ordinances and county ordinances for that area.
If your dog was injured or killed by a dog or a person, see When a Dog Is Injured or Killed.