Yes, most people do forget their dreams. Scientists have proven that if people didn't dream every night, they would go clinically mad.
Also, even on the dreams you do remember, 80 to 90% of it is lost after the first 5 minutes of waking up.
Even unborn babies dream, as thanks to new 3 and 4D scanning, they have discovered that unborn babies have rapid eye movement sleep (and we dream during rapid eye movement).