You're right, LilRed, dreams are fascinating. Neurologically speaking, one theory which I favor is that they serve a kind of safety-valve function that makes it possible for the subconscious mind to express itself.
Sleep is the only form of unconsciousness that does not indicate a serious health condition. REM, or rapid-eye-movement sleep, is when most dreaming occurs. During REM sleep, oxygen absorption is actually higher than it is in a fully alert individual; blood pressure, heart rate, respiration are all elevated, while digestive motility is down. In other words, you have a sympathetic nervous system reaction during the REM stage of sleep. REM brain waves are also typical of alertness, not unconsciousness.
It has been noted that lack of REM sleep has ill effects on people; we need to dream. This is why I believe the safety-valve idea has merit; dreams give our neural pathways outlets for excess energy.
It's also fascinating to me that during REM sleep, a temporary paralysis is in place throughout the muscles and skeleton; this prevents the acting out of dreams.
Someone is probably now thinking, 'Aha! What about sleepwalking?' Well, this is another fascinating facet of dreaming and sleep: although most dreams occur in REM stage sleep, nightmares and night terrors tend to occur during NREM stages 3 and 4; sleepwalking often coincides with night terrors, and nightmares are statistically speaking acted out more often than pleasant dreams.
This is a brief summary of what neurological science can say about dreams. As to what they mean, it is my opinion that depends on the dream. I've had dreams that were messages from my subconscious: you know, you dream about frantically looking for a bathroom in an unfamiliar place, and can't get to one; so you wake up and find you really need to find a bathroom (but presumably it's easier than in the dream!). Or I've dreamt about things I neglected to do, and that served me as a reminder the next day.
I've also had dreams that were nonsense, and dreams that were in my opinion a form of soul travel, or shamanic journeying. This is all outside the realm of science, which can only penetrate neurological mysteries so far at this point. But of course it's discovering more every day!