The idea of dominance can be very confusing, because there are so many different contexts in which the term is used, and it is frequently mis-used. When discussing domesticated dogs, the idea of dominance can be boiled down to a simple concept: I want the best spot!
The concept of dominance is actually quite fluid and is very context specific. Make a list of the resources available in your home, including food, water, space, toys, etc...all that stuff that your dog needs and likes. Now create a list of the dogs (and even the people) in your home. Add in the different locations and situations in which your dog encounters those resources. Now multiply that all out. Wow - that's more math than I can do! That's how many different responses your dog may have to competition for specific resources. And that's what dominance is really about - who controls the resources.
Dominance isn't typically about conflict. Consider the dog who through stealth aquires the abandoned bone. No conflict involved, but the bone resource is now controlled by the stealthy dog. So the next time someone tells you to assert dominance over your dogs, there's a simple answer. "Sure - I do it daily. Who do you think buys the kibble at my house? Opens the back door for potty breaks and play sessions? Distributes treats and pets for good behavior? That's me!" The beautiful thing about being human and posessed of both an opposable thumb (handy for opening those doors!) and an ability to reason, is that we humans can easily control the resources without creating conflict.
Not sure you're up for the challenge of a conflict-free household? The Austin Dog Trainer can help you to create a positive relationship with your dog via our dog and puppy training classes and private behavior modification lessons.