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Personally I think the term lucid is much overemphasized. The whole idea is simple - a dream is lucid if you become aware in the dream that you are dreaming. Some people also associate lucid dreams with a control of what happens over the dream. You can perform magical tricks during the dream, you can fly - that kind of thing. Personally I would define a lucid dream as a dream where you are aware you are dreaming and also that the dream is extremely vivid.
Much of the obsession with lucid dreams revolves around the invention of dream machines. These are masks which fit over the face during sleep. They recognize when you are dreaming because the eyes move in REM. Once this dreaming starts the dream machines set off lights within the mask and also sounds which are aimed at drawing attention to the fact that you are dreaming. And with training you can develop the ability to wake up during the dream and even control it.
There are some practical applications of this. For instance people can practice certain tasks in life with this...such as driving, speaking or even conducting an orchestra. It is argued that the neuronal patterns during REM are the same as during the actual activity. You can actually practice a task.
Secondly lucid dreamers also claim to solve nightmares with this technique. They can be chased by a pack of dogs, and because they are dreaming lucidly they simply realize that the dogs are symbols of their imagination and let themselves be devoured.
Personally I think both are fairly dubious. Firstly people were inspired in dreams long before the super nova dream machines were invented. Just concentrating on wanting a solution to a problem will set the mind to a particular intellectual task during sleep. Secondly if a person is enlightened enough to use a dream machine to induce lucid dreaming then its unlikely they would allow themselves to be so scared by nightmares.
If you do not use a dream machine then lucid dreams are most common when you wake up early - then stay awake for a short while - then fall back to sleep. Interrupted sleep is often followed by lucid dreams. Lucid dreams can also happen immediately after you wake up. You wake up then lapse back into sleep immediately..which is often quite lucid.
Lucid dreaming then is best described as dreams which are experienced during disruption of sleep. Some believe that lucid dreams are a special and much more important dream than normal dreams. I am personally not convinced. To me I am quite happy for dreams to come along when they feel like it. And lucid dreams involve interrupting your sleep which is not very appealing to me.