Measurements of sleeping subjects show that the brain is very active during sleep. While the body rests, the brain re-energizes itself. It's not just the quantity but the quality of sleep that can determine whether you have a sleep debt. Research shows that most people get a whole hour of quality REM sleep between the seventh and eighth hour of sleep. However, if you only sleep six hours, you are missing that last hour of REM sleep completely.
Lack of sleep can make it difficult for a person to perform normal activities.
Recent studies suggest that brain activity either decreases or becomes less efficient with lack of sleep. Specifically, research subjects show less ability to memorize and recall words or complete simple math problems after 35 hours without sleep. Interestingly, maps of blood flow and electrical activity in the brain show that the tired brain tries to shift the processing for verbal tasks from one area to a bunch of areas in order to compensate. It doesn't work all that well. However, no such compensation activity is observed in a tired brain trying to do math problems.