It all depends which kind of animals you're talking about.
For instance, I don't think you can put oysters and dolphins in the same category.

Some "animals" seem to have cognitive capacities very close to us, and possibly even superior in many aspects. An elephant has a brain three times larger than ours, a larger neocortex, more mirror neurons and spindle cells (those that seem to be involved into what we call our "identity" or our awareness feeling), a better memory, and can succeed to counting tests way better than humans.