If you're familiar with the sleep cycle (i.e. the 5 stages of sleep), then you'd likely also know that the various proportions of time spent in each stage changes with each cycle in a predictable way, such that for each succeeding cycle, less time is spent in deep sleep and more in REM. So what happens if one is sleep deprived? Well, fewer cycles of sleep are achieved, and hence the overall proportion of time spent in various stages of sleep is different as compared to a "well rested" sleep. Most notably, REM is significantly reduced since REM time is very short in the first 2-3 cycles of sleep. However, continue this, and it appears that REM starts to fight back and take back some time off the remaining sleep that is available.
This is what I've experienced so far (across the years, and fresh from recent 2 weeks or so). It would appear that REM is highly important, and that the mind/brain is willing to sacrifice the other stages of sleep in order to give REM more time. This happens at around 2-3 days of deprived sleep (deprived being less by ~2hours compared to normal). REM is easy to notice because it's during then that dreams (usually vivid) occur, and hence is a memorable time. (Everyone dreams. Don't think you're the exception.)
I've read about polyphasic sleep, and a blog of a person going through it for about 2 years. Apparently, REM is all they ever go through since the periods of sleep are so short, and it is enough to recharge, revitalise, and ready the mind and body to carry on. You can easily see that if REM fights back to always retain itself no matter how short the sleep, then for polyphasic sleep, it naturally consumes everything, basically. There's not been enough study on the long term (read, years) effects of lacking deep sleep though, so I'm unwilling to try this, but there you go. Dreaming (REM sleep) is terribly important, and you'd probably notice this yourself, if you're an avid oneironaut.