People who claim to dream more than others usually awaken during the REM stage of sleep, making it easier to remember their dreams. Generally, all people dream about the same amount each night.
The majority of dreaming takes place in the REM stage of sleep and when someone awakens during this stage, they often remember their dreams very vividly. This can make it seem as though a person dreams more than someone else.
People have five stages of sleep. The first four are called NREM stages and the fifth, which is the stage when most dreaming occurs, is the REM or "rapid eye movement" stage.