In this philosophical sense of the word "conscious," we are conscious even when we are dreaming; we are conscious of what’s happening in the dream. But sleep researchers believe there is a sleep stage that happens, called "deep sleep," in which apparently we are not conscious of anything in any sense. No mental processes that involve consciousness in an ordinary or a philosophical sense are going on. So deep, dreamless sleep would be an instance in which one is alive, and one's brain is functioning, but there are no mental events occurring in which there is any element of consciousness.
There has been some debate about the following question: Must one be conscious, in the philosophical sense, whenever a mental event occurs? For example, is it possible to have a pain that one does not feel? Some people think not; they think that, in order for something to be a pain, one has to feel it or be aware of it. Similarly, if anything is a thought, then one has to be aware of that of which one is thinking (indeed, that seems nearly a tautology); if there is no consciousness, no awareness, of anything at all, then one is not thinking. Philosophers ask: Do MentalEvents? necessarily involve consciousness?
Suppose we answer "No." Then of course what we’d be saying is that there are some MentalEvents that do not include an element of consciousness. These events are going on even though we aren’t aware of them. In other words, part of the mind is unconscious. Put like that, it's easy to recognize why the thesis is compelling. Psychologists of course believe that many cognitive processes are unconscious; we are aware of only some of the stuff that’s going on in our minds. That is a fairly recent view, something that was made popular only after Freud.