In REM Sleep

Animals, fury, feathery or scaly, have similar sleep patterns to humans. (Seeing as humans are animals, this makes sense.) It is quite probable that many animals even dream- particularly dogs and cats.

In many animals, the REM stages of sleep can be easily detected by simply watching them. Some animals sleep with their eyes open. When they do this, their eyes often roll back into their heads and you can easily see their eyes twitching. Even if they sleep with their eyes closed, if you watch closely, you can see that their eyes begin twitching and rolling around. For the most part, it is not known for sure whether or not animals actually dream. Scientists do believe that they do dream; it is thought that animals’ dreams express wish fulfillment as Freud’s theory suggests about humans.

Through an experiment gone awry, scientists at the University of Pittsburgh accidentally found out that monkeys do in fact dream during REM sleep stages.
Dogs, it is thought, dream about hunting. The common belief that they dream about chasing rabbits is supported by their behavior during REM sleep. One can observe a dog’s ears turn back, its neck stretch out, its breathing shows strain of exertion, and its feet twitch and churn as if running. These are all behaviors exhibited when a dog actually chases a real rabbit.
Cats sometimes act as if hunting a mouse while they are asleep. A sleeping cat that tenses its body and waves its tail slowly looks similar to an awake cat that is getting ready to pounce.