I feel I must point out that nREM sleep is not dreamless sleep, but less vivid dream sleep.
The “conscious” mind can be defined many ways. Depending on how you do define it, you will find that it may or may not be resting.
If by conscious mind you refer to the brain as a whole, then the brain never really sleeps. It remains active to some extent. During nREM sleep, the brain is less active than during waking life, but it is still active. The brainwave patterns are slow, controlled, and very cyclic. Thus, the brain at this point is kind of taking a rest, but it is not really resting. During REM sleep, the brain is as active and alive as if you were really awake, but sleep paralysis prevents you from moving about.
This whole pattern can be likened to a marathon runner: the runner can be very active and running quite quickly (awake like during REM sleep) or he can be taking it easy and moving at a controlled smooth pace (nREM sleep). The runner (or conscious mind) alternates between these two states on his run (sleep) to make the most efficient method of travel (runner only ).
If when you refer to the conscious mind as the part of the brain that involves self-awareness and self-reflection (the right dorsolateral prefrontal cortex) than during non-lucid dreams, that part of the brain is largely inactive the entire time (nREM and REM sleep), and can said to be “sleeping”. Thus, if this is your definition, the conscious mind does sleep when you go to sleep.
However… it doesn’t necessarily have to as during lucid dreams this part of the brain “comes slightly alive”. Thus, when you have an LD, that part of the brain activates during sleep, and, to continue with the previous metaphor, gets in on the action.
Many, MANY studies have been done on the purposes of dreams, and one conclusion in particular that I recall made the most sense to me. (Here’s the link to the video in case you wish to view it for yourself.)
Basically (summing up from the link above) the brain during nREM sleep is re-enacting known information to determine what is important and what is not. During REM sleep, the brain is attempting to take the known information and apply it to randomly generated new concepts to determine future usefulness. This can easily be explained by likening the brain to a training simulator in which it attempts to prepare us for the future ahead.
During this time, though, that particular part of the brain involved in self-awareness and reflection is basically not involved in the action. It is theorized that this is because that part of the brain inhibits the intuitive approach that the mind takes during our sleep.
Hope this has been helpful and / or useful. And do note that this is all my opinion influenced by the opinions below.
SOURCES:
[spoiler]NOVA - What Are Dreams? (Video)
Sci Show: Dreams (Video) [this one eludes me for the moment… sorry… ]
Seat of Self Awareness (Web Page)[/spoiler]