From "the leader of the pack" to "the top dog," plenty of simplistic metaphors come from the canine world. But relationships between canines can be pretty complex, beginning with the very first meeting. Like most animals who live in groups, dogs establish their own social structure, sometimes called a dominance hierarchy. This dominance hierarchy serves to maintain order, reduce conflict and promote cooperation among pack members. Dogs also establish territories, which they may defend against intruders or rivals. Obviously, dogs' social and territorial nature affects their behavior whenever a new dog is introduced to the household.
If you have more than one resident dog in your household, it may be best to introduce the resident dogs to the new dog one at a time. Two or more resident dogs may have a tendency to "gang up" on the newcomer.
It is important to support the dominant dog in your household, even if that turns out to be the newcomer. This may mean, for example, allowing the dominant dog to claim a favored sleeping spot as his or to have access to a desirable toy. Trying to impose your preference for which dog should be dominant can confuse the dogs and create further problems.
Puppies usually pester adult dogs unmercifully. Before the age of four months, puppies may not recognize subtle body postures from adult dogs signaling that they've had enough. Well-socialized adult dogs with good temperaments may set limits with puppies with a warning growl or snarl. These behaviors are normal and should be allowed. Adult dogs who aren't well-socialized, or who have a history of fighting with other dogs, may attempt to set limits with more aggressive behaviors, such as biting, which could harm the puppy. For this reason, a puppy shouldn't be left alone with an adult dog until you're confident the puppy isn't in any danger. Be sure to give the adult dog some quiet time away from the puppy, and some extra individual attention as well.
If the introductions don't go smoothly, contact a professional animal behaviorist immediately. Dogs can be severely injured in fights, and the longer the problem continues, the harder it can be to resolve. Punishment won't work, and could make things worse. Fortunately, most conflicts between dogs in the same family can be resolved with professional guidance.