as we all know, the moon is spherical. or if we see it in a 2-dimentional view it is circular. during full moon, the moon will look like perfect circular, reflecting the sun light to the earth. but during its crescent shape, the moon looks like ... erm, a crescent of cos, 哈哈 (how to describe the shape =P). it is like less than half circular. it becomes like dat bcos the other part of it is being blocked by the earth. so what we see on the moon of its "dark side" is actually earth's shadow.
so basically the moon still in round shape, but we see only the crescent part of it being lighted up by the sun. the rest should be black in shadow. but in that 王心凌 -- 月光 song, the dark side of the moon seems like dissapearing, showing that the moon is only within that crescent shape, and let the clouds behind the suppose-to-be-shadow of the moon, appears!! this is actually one of the misconception in depicting moon's image to the people.
similar analogy would be having a blasting sound if there is any explotion occur in outer space, which actually it will never happen. outer space is basically vacuum with no air at all. sound needs medium to travel to be heard by us.