Obviously, we cannot know for sure what dogs dream about. However, a number of scientific studies seem to all be in agreement that dogs dream of "dog things". In other words, they dream of what really happens in their lives, such as chasing the mailman or the cat, digging holes, and protecting against burglars. 
Structurally, a dog's brain is similar to a human's brain; when they dream, their brain waves have the same patterns of their two-legged counterparts. Their electrical activity is the same as a human's, as well. You can tell a dog is dreaming when their leg twitches, when they quiver, and even when they moan in their sleep.