Everyone dreams, but some people sleep so heavily that they don't remember their dreams. The dreams that are most likely to be remembered are those when you are in light sleep and about to wake. Other times when you are abruptly awakened from deep REM sleep. I tend to sleep very lightly, waking up frequently, therefor I have long lists of dreams every night or the ability to continue a single dream if I prefer.
I'm the same way...er sorta. I can tell I have some crazy thoughts and crazy dreams while I'm sleeping, but the moment I wake up they're instantly gone and there's nothing that can bring them back (maybe a slight case of deja vu?) I consider it lucky if I have a dream I can clearly remember.