From what I have read, our dogs dream in just about the same way that people dream. They go into a deep REM sleep, and then start dreaming. We often dream about things that happened to us during the day, or even about television programs or movies we watched that night; and our dogs dream about the things they experienced as well. Sometimes, it seems like our dog might be having a nightmare, and then we have to decide whether or not to wake them up out of the dream. I know that I am often really disoriented if I wake up out of a very vivid dream, but if it is a nightmare, then I am glad that I finally woke up !
The experts suggest that it is best not to wake your dog up when he is dreaming, and usually I don't bother them, but if they sounds really agitated and like they are having a nightmare, then I will speak softly to them, and at least try and reassure the dog that everything is still okay.