We all dream. There's a lot that we can't remember though. If you want to remember your dreams, try to write them down upon awakening. In the first 5 minutes of waking up you'll start to lose about 50% of your dream memories.
I too, haven't dreamt for a long time. Nothing like
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japsander
's 20 years... but, it was a little while. Only recently did I start dreaming again. Starting to recall even ancient dreams 20 years ago randomly.
One dream I had every night for probably 10-12 years as a kid was kind of like the introduction to REM for me. I would see a bright yellow room, with only a stove in it. The stove would be heating up, and the door to the oven would be opening and closing repeatedly with thick, black smoke billowing out of it. And the room would fill with smoke until I couldn't see anything and it was just blackness. That's when (over time) I noticed I fell into my deep sleep.
I had a dream a couple weeks ago about my ex. Her best friend ran up to me and hugged me. She said "thanks for giving Ana that rabbit; she's so happy and loves it" I told her about it, and she said that the night before had a dream about meeting up with me, and I gave her a fluffy toy rabbit.
Dreams can be all sorts of things. I think/believe they're our subconscious trying to make sense and analyze everything going on, solving riddles, questions :P Learning us as we sleep.
They also creep the hell out of me. I've never had a nightmare as I've always been able to control my dreams. Had seen some weird stuff though.
It's also said that all of the faces you see in your dreams are people you've met/seen throughout your life. All familiar faces (to a degree); people you've seen walking around, family or friends, that girl in the porno mag you jack to :P