As the Moon moves in its orbit around the Sun, our view of the side illuminated by the Sun changes.

At full Moon, the moon rises at sunset and sets at sunrise. Excuse the misspelling in the figure; someday we will fix it!:

At 3rd quarter, the moon rises at midnight and sets at noon. Then we see only a crescent.

At new, the moon rises at sunrise and sets at sunset, and we don't see any of the illuminated side! Then we see only a crescent, but compare this to the waning crescent: what's different? Why?