Yes, dogs do dream.
We have good dreams and we have bad dreams.
The other day I was laying down wagging my tail and my owner realized that I was having a good dream. When I have nightmares I will shake and even try to bark or growl. If my owner catches on she usually tries to wake me.
What are my dreams about? Well, that's hard to say. I usually can't remember them afterwards. When it's good I may be playing in the snow, with a friend, or getting a really good treat - like a giant raw hide. If it's bad, I may be dreaming about a bigger dog or being dropped at the pound where my owners got me originally. I don't EVER want to go back.
Either way I do dream a lot since I consider myself a professional sleeper. I practice often.