Usually I wake up knowing that I've dreamt something -- usually something incoherent and random -- but I seem to forget exactly what the dream was about mere moments before my eyes open. From what I've read, this is how our "system" is supposed to work.
But there's a special kind of dream that always "sticks out" and I can recall it very vividly even years after. I believe they're called lucid dreams, and the main thing about them is that you're somehow "awake and aware" inside the dream: You seem to vividly "experience" what you dream, all the while knowing that it is a dream. A rare, strange yet thrilling experience to say the least...!
On one occation I dreamed about climbing up some girder mast, causing it to first sway off to a side because of my weight, then "swiping" back, hurling me through the air, across the football fields of a local stadium, and ... then I realized this would be a harsh "landing", so I decided to wake up from the dream before impact. The view over the town from the apex of my "flight" was neat, though...
On another occation I dreamed I went to the local mall on bicycle along with a then-friend of mine. After the visit at the mall, my bike had been stolen, but hey, this is just a dream; I knew that in reality my bike was safely parked back home in our garage, so "no worries mate!"...