Delta waves are comparatively the strongest signals that your brain produces but also the slowest. If the beta waves of a highly active mind are like a bunch of rapid, tiny ripples in a pond, delta waves are like the huge, slow wave of tides.

We never see a brain produce delta wave when it is awake… only in a very deep sleep. Since is it the only one that we can say that about, and we know that if you don’t get enough sleep it has a lot of negative physiological consequences (up to even death), some think that the brain NEEDS to enter this kind of state from time to time.

But nobody is really sure about that. Even the question of why we need to sleep at all is up in the air and has been for a long time (one neurologist pointed out that anyone who comes up with a provable idea will probably get a Nobel Prize for it).

What we can say with some certainty is that if someone’s brain doesn’t enter a delta state, they’re probably not getting very good sleep at all. Theta is associated with dreaming (either day-dreaming or sleep-dreaming), but sleeping people normally only do that every 90 minutes or so.

Link below for more detailed information. Hope that helps!
References :
http://www.web-us.com/brainwavesfunction.htm

Delta activity is characterized by frequencies under 3 Hz and is absent in awake healthy adults, but is physiological and normal in awake children under the age of 13. Delta waves are also naturally present in stage three and four of sleep (deep sleep) but not in stages 1, 2, and rapid eye movement (REM) of sleep. Finally, delta rhythm can be observed in cases of brain injury and comatic patients.

Human non-rapid eye movement (NREM) sleep is divided in the categories 2, 3, and 4
References :
wikipedia