The amazing world of dreams... I could go on and on endlessly about this topic. At a basic level they're just your subconscious working images on you from often subliminal things. At a deeper level, they are a window in to your soul and are so much more powerful than they seem.
I'll get to that later though. Firstly, I also enjoy puzzling out the mechanics to abstract stuff and I also have no idea how dreams work fully... but I don't think anyone does. It's like how bees shouldn't fly or how planes only fly in theory (because I always have an image of a famous mathematician on a plane suddenly exclaiming "Hang on, according to this planes shouldn't be able to fly!" and all planes suddenly crashing to earth!).
On to the main thing... Apparently you have something like 1000 dreams per night on average but you only remember a couple of them the next morning, even fewer by the end of that day and so on. Usually a 'dream' is actually several dreams that run in to each other (ever had a dream that makes perfect sense to you at the time, though when you try to explain it later you go "Oh, well we were in the house... then suddenly the walls turned to bamboo and we were in this huge forest!" etc etc... I imagine that would be one dream running in to another with similar themes). Still, whether broken up or not, they make perfect sense to you and can impact upon you. Also considering that your brain runs at a different speed, it may seem like a dream goes on for hours even though you've only been asleep for a few minutes in reality.
I definitely have powerful dreams. I live next to the water, so when I had a dream one night and saw a boat there, that was nothing special. I jumped on to this dingy which looked very much like my father's and there was a trapdoor at the bottom of the front. Opening the trapdoor, I saw a ladder that went down a bit (which I know makes no sense as the boat was on the water but stay with me here). I came down in to this house. It was all wooden and beautiful. It was adorned with hundreds of items, coats on racks, glass cases with rare items inside them, beautiful rugs on the floor... I met a person inside that house too, he was very pleasant but I can't remember much more about him. There was a large window (which was made up of many smaller windows, like the type you see on ships from the 1700's - kind of thing, which looked out over the sea and the waves. It had many corridors, all filled with these wonderful items, old golden spy-glasses, things you might wear in a play (drama costumes)... I spent ages exploring this beautiful wooden house under the family boat. I felt entirely safe and entirely at home while doing it. I never finished exploring the house because I woke up before I could keep on going.
When I woke up, I asked my mother about it. A person who's done some research in to dreams, she told me that houses often represent you. The items inside that were beautiful, she said, meant I was a beautiful person. We went through for ages talking about what the items meant... Nowadays I take comfort in that house, the dream. I don't fully know what it meant but I still remember it so vividly... it's important to me.
So yes, I do have powerful dreams. Other dreams include things like deja vu (which everyone gets... Someone told me that scientifically, it's more a case of you see something that you've seen before waking and think that was a dream. I disagree as i've had deja vu with places i've never seen before, to which the person replied "well, that's the best science can say for now anyway!
"), painful dreams (what's your worst? I've had my stomach ripped open by a monster in one of those dreams... ughhhhh...), lots of others.
There are dreams where I know i'm in a dream. I recall an instance where I was on a yellow taxi driving past houses in Britain (trust me, It was Britain. I've never been there personally but that was Britain.). I knew it was a dream, the first thing I did was call out to everyone that it was a dream. no-one responded, so the next thing I did was smash my head against the top of the taxi. Fourth or fifth hit and I woke up.
Others are where you subconsciously know you're in a dream. I was hiding on my balcony in a dream once and when the 'bad guy' saw me I decided it would be best to jump off head first on to the brick flood wall that was there (since the house that i used to live in was next to a creek, it had a flood wall), knowing that it would get me away from him.
You can have dreams where you're controlling what's going on, both when you know and don't know. Those are most fun when you know. I remember my first one, I was crawling through pipes but looking at it as though it was a puzzle game (so I could see the whole maze and me in it). There was a lolly in front of me and I remember eating it and enjoying it, so I made another and ate that one, and so on until the dream changed/I woke up (not sure which happened in that instance). Another early one was where I made myself fly and everything turned in to a spyro level (because my brother had been playing spyro and as a young kid I was always watching him and playing it myself.).
On that note, you can learn to control your dreams every time. My brother learnt to do it and did it for years. The trick is remembering something before you go to sleep, the better you know it, the more likely you'll recognize it in dream. I always used a metal vase that had significance to me when I was doing it. Basically, the trick works in that if you see the object out of place in the dream (for instance, if you memorized the moon and saw the moon in the middle of a street, you'd know you were dreaming.). That would alert you to the fact that you were dreaming. From there, you could begin to control it how you wished.
A very powerful type of dream is what my family calls a 'psychic dream'. My mother was in Sydney and he grandmother lived in Melbourne, many years ago. One night my mother had a dream where she was at the top of a skyscraper and at the edge her grandmother was clinging to the side. My mum rushed to her and their fingers brushed, before her grandmother fell in to the abyss below. The next morning my mother woke up and immediately got in to her car. She drove all the way to melbourne without warning. When she got there, she was directed to the hospital by family members and was told when she got to the hospital that her grandmother had literally passed away 30 minutes before she arrived. And she drove to melbourne without any provocation, no calls from family or anything. This is exactly the type of thing i'm talking about, that dreams can really influence you. if you wake up one morning and drive hundreds of kilometres because of a dream, then yes, i'd say dreams like that are extremely powerful.
I think the most powerful and best dream I have ever had was when I was about 7 or 8 years old. I've never had this happen again but I want it to. It's called Astral Projection. Most people who've had it (and I kind of agree with them) say it's the soul coming out of the body and going wherever you want it to go.
I had fallen asleep on the couch that night after coming home late from some kind of social function. Next thing I knew, I was watching myself sleeping from about 7 or 8 feet above the ground. I remember looking so peaceful. My sister had an instance where she did it as a young girl. She heard her parents in the other room talking... next morning she went and repeated every word of the conversation to them and found out they had had that exact conversation that night. A brother of a friend of mine projected when he was little and saw a house a few blocks away burn down. Next morning, turned out that exact house had burnt down that night.
Eerie but fun. It seems to me that it's not so much of a dream... your soul is actually moving through the real world, flying, watching, etc... I've promised myself that if I ever do that again and realise it, i'm going to travel a bit and see what I can see. It would be very cool to see a bit of China without ever actually going there in person!
So yeah, that's a bit of my experience with dreams. I'm not overly spiritual or anything like that, these are just experiences that I or family members have had and how they've impacted upon us.
I think that, whether we realise it or not, we are often very influenced by our dreams. They give us such insight in to ourselves and the world around us that we take things from them and apply that in real life, whether we know it (psychic dreams, dreams that reveal things about ourselves, etc.) or not (deja vu, etc.).
Sorry for the wall of text! That's basically what I wanted to say though.