I think it’s pretty logical for everybody that our brain doesn’t simply shut down during sleep. Had it not had to regulate the vegetative processes, maybe it should have shut down, but since it really has to control the heart rate, the digestive system and all the other (non-autonomic and vegetative) functions, it has to stay alert.
However, the brain itself has several resting periods, regulated per se by the sleep waves. Usually, in the first hour, all the processes are calmed. The brain controls the sphincters and the vital functions, the only difference being that they are slow-downed pretty much. As a result, the temperature of the body, the activity of the digestive symptoms and the heart rate (implicitly the blood pressure) are decreased.
After the first hour of sleep, the brain enters in a so called REM (acronym for rapid eye movements) sleep. It’s a more active form of sleeping in which the brain doesn’t necessarily rest. Some neuroscientists say about the REM sleep that it opens for the brain a door to see the big picture of one’s lives, although this is not commonly approved by the high medical society. In this phase, out skeletal muscles are somewhat paralyzed as a form of protection against falling out of bed and the eyes move rapidly. Usually, this is when active dreaming takes place. If waken up, people can’t approximate the hour, but have memory parts of the dreams.
The non-REM sleep is the most resting period for the human brain. It alternates with the REM phases several times a night on different patterns, depending of the age. For example, infants have way more non-REM phases than the regular adult since they sleep for about 18 hours a day, while old people have few non-REM sleep phases a night, since the average rest time for a senior is about 5 hours a night or less.