Compared to humans, much of a four legger's brain is devoted to nasal memory (related to their sensory input — a nose close to the ground). A dog can identify whether pairs of humans are identical twins or not by scent alone.
Their memories, thoughts and reveries may be rich and difficult for us to fully understand.
“The past still lives in us …[it] has made us what we are and is
remaking us every moment! … An hour is not merely an hour! … It is a
vase filled with perfumes, sounds, places and climates! So we hold
within us a treasure of impressions, clustered in small knots, each
with a flavor of its own, formed from our own experiences, that become
certain moments of our past.”
— Marcel Proust, À la recherche du temps perdu
Some tidbits from a DARPA “artificial nose” researcher that I met at the IBM Institute on Cognitive Computing: Compared to humans, the dog has a much larger olfactory epithelium (the sensory receptor sheet) with many-fold more receptors. Also, dogs (and many other mammals) have more types of sensory receptors, and so they encode odorants in a higher-dimensional representation (reminding me of vision in birds). Humans have about 350-400 distinct receptor types, whereas mice have 1000. Many of these receptors are co-activated by the same odorant, so biological olfaction is inherently a high dimensional representation system.
The olfactory system is a much more direct activation path for stored memories, at a much deeper and more multidimensional level, than the other senses, consistent with its primitive evolutionary role in memory and association. The electrical signals from the olfactory tract go to the limbic system, the most ancient part of the brain — the hotbed of our most instinctual and primitive emotions.