[This post is human-centric. No big surprise there, since a dog clearly did not write this. Bias is ok as long as we work in awareness of it. For a more accurate insight into the process of domestication check out this article on dog domestication and borrow The Botany of Desire by Michael Pollan from your local library.]
Whether we domesticated them for food, labor, defense or companionship (or a combination of all) we have shaped not just the physical attributes but the cognitive functions of domesticated species.
We have made dogs and cats docile, perpetually juvenile, compared to their wild counterparts. This has even extended vocal communication in cats. Feral adult cats and wild cats rarely vocalize while their domestic relations mew dependently for every meal. Dogs remain playful long after physical maturity, extending their learning as well as their adolescence.
Darwin aptly articulated the physical variations human selection of plants and animals creates among domesticated species. The same can be said of the cognitive capacity and protomemes of domestic animals compared to variations within their species and in comparison to their wild relatives. A Pit Bull's response to physical threat is quite different from a Sheltie's. The adult Wolf's means of acquiring dinner are quite different from the Papillon.
Cats have retained more independence as their main function was pest control and they needed little encouragement to fill that niche. Dogs on the other hand depended on reading social signals and moods of humans. Because of this dogs developed an ability to read human facial expressions, and respond to verbal and non-verbal cues (which most cats, as any roommate of one will know, feel too intelligent to do on any reliable basis). Dogs even developed the ability to point (for hunting purposes) and to decode human pointing (something most creatures, even those closely related to us, cannot do).
This shaping of cognitive and emotional function extends to farm animals as well and explains why animals in confined, industrial environments respond in ways similar to human psychological disorders. We shaped them to function in a pastoral environment, complete with social stimuli, human husbandry and natural cycles, then rapidly changed that environment. Since we control the breeding, even after the species has lost its ability to mate independently, such as with Angus cows (where natural mate selection is not permitted) and Domesticated turkeys (where the birds were bred to have so much meat they are physically incapable of breeding), these animals have lost all ability to adapt or go extinct, caught in a vicious cycle of human manufacture, driven by markets rather than gene-environment interplay.
But there is another brain/body which we have domesticated with even more complex outcomes: Our own.